473 



916a 
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E STRENGTH AND STIFFNESS OF STEEL UNDER 
BIAXIAL LOADING 



BY 



ALBERT JOHN BECKER 

B. S. in M. E. University of Michigan, 1903 

M. E. University of Michigan, 1907 



THESIS 



Submitted in Partial Fulfillment of the Requirements for the 



Degree of 

DOCTOR OF PHILOSOPHY 
IN ENGINEERING 

IN 

THE GRADUATE SCHOOL 

OF THE 

UNIVERSITY OF ILLINOIS 
1915 



THE STRENGTH AND STIFFNESS OF STEEL UNDER 

BI-AXIAL LOADING 



BY 



ALBERT JOHN BECKER 
B. S. in M. E. University of Michigan, 1903 
M. E. University of Michigan, 1907 



THESIS 



Submitted in Partial Fulfillment of the Requirements for the 

Degree of 

DOCTOR OF PHILOSOPHY 
IN ENGINEERING 

IN. 

THE GRADUATE SCHOOL 

OF THE 

UNIVERSITY OF ILLINOIS 
1915 






m 



7- 27^0 



university of illinois 
Engineering Experiment Station 



'O 



Bulletin No. 85 April, 1916 

THE STRENGTH AND STIFFNESS OF STEEL UNDER 
BIAXIAL LOADING.* 

By x\lbert J. Becker, 

Professor of Applied Mathematics ix Uxiversity of North 

Dakota,, and For:merly Graduate Student in University 

OF Illinois. 

CONTENTS. 

I. Introduction 

Page 

1. Scope of Investigation 5 

2. Acknowledgment 5 

3. General Statement 6 

4. Combined Stress 1^ 

II. Theories of the Strength of Materials Under Combined 

Stress 

5. The Six Theories 8 

6. The Maximum Strain Theory 8 

7. The Maximum Stress Theoiy 10 

8. The Maximum Shear Theory * ... 10 

9. The Internal Friction Theory 12 

10. Mohrs Theory 13 

11. Wehage's Theory 13 

12. Graphical Presentation of Three Theories 14 

*Tliis Ijulletin embodies the principal data of the thesis of Albert John Becker, presented 
in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineer- 
ing in the Graduate School of the University of Illinois, June, 1915, together with further 
experimental data taken by the author to extend the investigation. 



/i CONTENTS 

III. Experimental Work 

13. Form of Specimen 16 

14. Cross-bending Tests .19 

15. Tube Specimens 19 

16. Preparation of the Tubes 22 

17. Determination of the Thickness of Tube Walls 24 

18. Method of Testing 25 

19. Character and Sequence of the Tests 27 

20. Test Operations 29 

21. Diagrams and Tables 31 



IV. Discussion of Eesults 

22. The Criterion of Strength 31 

23. Strength 35 

24. Stiffness 45 

25. Correlation of the Eesults of Other Investigations .... 49 

26. Summary and Conclusions 54 



Appendix I, Bibliography 
Appendix II, Mathematical Treatment 

1. Stresses and Strains 58 

2. Stresses and Strains in a Thin Tube 61 



CONTENTS 3 

LIST OF TABLES 

1. Outline of Test Specimens and Tests 28 

2. Data of Tubes 30 

3. Test Data of Tube No. 4, Series 1 32 



LIST OF FIGURES 

1. Illustration of Stresses Produced on Oblique Planes in a Bar Subjected 

to Torsion 6 

2. Graphical Representation of Stresses According to the Alaximum Strain 

Theory 14 

3. Graphical Representation of Stresses According to the ^Maximum Stress 

Theory 15 

4. Graphical Representation of Stresses According to the ^Maximum Shear 

Theory 16 

5. View Showing Cross-Bending Test Specimen Under Load 18 

6. View Showing Compression Side of Cross-Bending Test Specimen 

After Test 20 

7. Dimensions of Tube Test Specimen 21 

8. Arrangement of Gage Holes on Tubes 21 

9. Location of Gage Lines on Developed Outer Surface of a Tube ... 22 

10. View of Apparatus Used in Measuring Thickness of Tube Walls . . 24 

11. View Showing Arrangement of Apparatus for Tension Test .... 26 

12. Stress-Strain Diagram Showing Johnson's Apparent Elastic Limit . . 33 

13. Stress-Strain Diagrams for Tube No. 3, Series 1. Ratio of Circumfer- 

ential to Axial Tension, 0.94 34 

14. Stress-Strain Diagrams for Tube No. 4, Series 1, Ratio of Circumfer- 

ential to Axial Tension, 0.69 34 

15. Stress-Strain Diagrams for Tube No. 2, Series 3. Ratio of Circumfer- 

ential Tension to Axial Compression, 0.20 36 

16. Stress-Strain Diagrams for Tube No. 5, Series 3. Ratio of Circumfer- 

ential to Axial Tension, 0.30 36 

17. Stress-Strain Diagrams Showing General Averages for Series 1. Ten- 

sion with Tension 37 

18. Stress-Strain Diagrams Showing General Averages for Series 2. Ten- 

sion with Tension 38 

19. Stress-Strain Diagrams Showing General Averages for Series 3. Com- 

pression with Tension 38 

20. Tension Tests of Small Specimens from Tubes of Series 1, 2 and 3. 

Results of Ten Tests for Each Series 39 



4: CONTENTS 

21. Torsion Test of Tube No. 6, Series 3 39 

22. Diagram Giving Yield-Point Stresses and Stress Ratios for Series 1 

and 2 40 

23. Diagram Giving Yield-Point Stresses and Stress Ratios for Series 3 . 42 

24. Representation of Yield-Point Strengths for Combined Stresses Accord- 

ing to the Maximum Stress Theory, the Maximum Strain Theory, 
and the Maximum Shear. Theory 44 

25. Relation Between Shearing Stresses Due to Torsion and the Tensile or 

Compressive Stresses Due to Axial Load or Bending 50 

26. Results of Tests by C. A. M. Smith 52 

27. Results of Tests* by E. L. Hancock 54 



THE STKENGTH AND STIFFNESS OF STEEL UNDER 
BIAXIAL LOADING. 

I. Introduction. 

1. Scope of Investigation. — The purpose of this investigation was 
to determine the laws governing the strength and stiffness of mild steel 
when subjected to combined stress produced by two tensions at right 
angles to each other or by a compression combined with a tension at 
right angles. In order to give a satisfactory basis for comparison of 
results, the plan of investigation provided that the ratio between the two 
stresses be kept constant throughout the test of a specimen, and J. B. 
Johnson's tangent method of determining the ^'yield point" or "apparent 
elastic limit" was selected. 

The specimens tested were drawn steel tubes of uniform size and 
practically of uniform thickness. These tubes were subjected to an 
axial load and to internal pressure. The only variable was the ratio of 
the circumferential stress to the axial stress. Comparison has been made 
only in the test results from sets of tubes cut from a single length of 
seamless drawn tubing. By means of strain gage readings a knowledge 
of the distribution of stress on the cross section was obtained ; no assump- 
tions wxre made except that of uniform distribution of the circumferential 
tensile stress throughout the thickness of the tube wall. 

The investigations of strength and of stiffness were carried on simul- 
taneously, but the results are discussed separately. The points investi- 
gated are : 

(a) The change of yield-point stress of the material with increas- 
ing ratios of circumferential tensile stress to axial tensile or compressive 
stress. 

(b) Stiffness of the material (strains accompanying stress) for 
increasing ratios of circumferential tensile stress to axial tensile or 
compressive stress. 

No discussion has been given of the engineering applications, for it 
is realized that while these applications are important, more work is 
needed to establish the conclusions reached. When this has been done 
and all the work has been correlated, it will be a simple matter to make 
an application of these principles to engineering design. 

2. Acknowledgment. — All the tests were made in the Laboratory 
of Applied Mechanics of the University of Illinois, under the supervision 
of Professors A. N. Talbot and H. F. Moore, to whom acknowledgment 
is made for their suggestions and criticisms and for the interest they 



6 ILLINOIS EXGINEEKIXG EXPEKI:MENT STATION 

have shown in the progress of the investigation. Acknowledgment i& 
also made to Mr. J. 0. Draffin, research fellow in the Engineering Ex- 
periment Station, for his assistance in the conduct of the various tests. 
It is also desired to make an acknowledgment to the Joint Committee- 
on Stresses in Railroad Track for the use of the new model 4-in. Berry 
Strain Gage. 

3. General Statement. — When a steel bar is tested in tension or 
compression, certain phenomena are observed which have been incor- 
porated as fundamental facts in the theories of the elastic behavior of 
bodies under stress. In such a test, both the strength and the stiffness 
of the material are observed, the former by noting the yield point and 
ultimate strength, the latter by observing the unit-strains corresponding 
to successive loads and computing the modulus of elasticity. Repeated 
experiments have sho^Ti that for material of the same composition and 




Fig. 1. Illustration of Stresses Produced on Oblique Planes in a Bar 

Subjected to Torsion. 



treatment, these results are practically constant and can be used as a 
basis of design. The strength of any material of construction cannot be 
determined by mathematical analysis, neither can its stiffness. Poisson's 
ratio, modulus of elasticity, yield point, and Hooke's law are experi- 
mental results. 

When an investigation of combined stress is attempted, there arises^ 
the question of the extent to which the calculations may be based upon 
the values obtained in the experiments in simple tension, compression, 
and shear. Constants determined by uni-directional loading cannot be 
indiscriminately applied to bi-directional loading. Theories have been 
evolved in which these constants are used by taking acconnt of the inter- 
action of the applied stresses. The analyses for these are correct from the 
mathematical standpoint, but the soundness of the basic assumptions carb 
be demonstrated only by experiment. 



BECKER STEEL UNDER BIAXIAL LOADING 7 

Different combinations of simple stresses are possible, and it may 
be expected that the same analysis will not apply to all combinations. 
The presence of shearing stress in a bar subjected to simple tension and 
the tensile and compressive stresses accompanying the shearing stress 
due to a torsional load indicate that the governing .conditions depend upon 
the relative strength of the material in shear, tension, and compression. 
A cast iron bar tested in torsion fails in tension on an oblique plane, 
because the tensile strength is less than the shearing strength. It is, 
therefore, logical to suppose that different stress combinations will pro- 
duce failures differing in character for different materials. 

4. Combined Stress. — Three types of stress applications are pos- 
sible, uni-directional or simple stress, bi-directional or biaxial, and 
tri-directional. The first is illustrated by a specimen subjected to tension 
or compression in an ordinary testing machine. Bi-directional or biaxial 
stress is the application of two stresses in the same plane acting in 
directions at right angles to each other. Tri-directional stress is the 
application of three stresses at right angles to each other. The condi- 
tion of biaxial stress is more important, from the point of view of the 
engineering applications, than that of three stresses at right angles to 
•each other. 

The possible combinations of biaxial stress are as follows: 

Tension with tension. 
Tension with compression. 
Compression with compression. 
Compression with tension. 
Shear (torsion) with tension. 
Shear (torsion) with compression. 

These may be divided into three classes, tension with tension and compres- 
sion with compression forming the first, tension with compression and 
•compression with tension the second, and the combination of either tension 
or compression with torsion forming the third. The third class includes 
also two special cases of the second class ; for a simple torque is equivalent 
to two equal principal stresses, one compression and the other tension, so 
that a torque combined with tension or compression can be reduced 
to the case of tension combined with compression or vice versa. This 
equivalence will readily be seen by considering a bar of circular cross- 
section subjected to torsion alone, Fig. 1. The stress on a plane at 
right angles to the bar is a pure shearing stress, depending in intensity 
upon the diameter of the bar and upon the torque. But this is not 
the only plane of stress. As in a bar in simple tension, so in this case 



8 ILLINOIS EXGINEERIXG EXPERIMENT STATION 

there are planes on which both tensile and shearing stresses occur; there 
are also planes upon which no shearing stresses occur. Referring to Fig. 

I, with the torque as shown by the arrow, the stress on the 45° plane CD 
is tension, and on plane AB at right angles to this plane, the stress i& 
compression. This is equivalent to a biaxial loading which develops a 
tensile and a compressive stress at right angles to each other and each 
equal to the shearing stress. It should be noted that there are stresses 
on oblique planes which may control the strength of the material. 

Applications of combined stress are to be found in the familiar ex- 
amples of the steam boiler for tension combined with tension, and of 
the crank shaft for tension or compression combined with torque. Bi- 
axial stresses occur in flat plates and in flat concrete slabs or girderless 
floors. 

II. TlIEORIES OF THE STRENGTH OF MATERIALS UnDER COMBINED 

Stress. 

5. Tlie Six Theories. — The mathematical discussion of stresses and 
strains in a thin tube under axial load and internal pressure is given 
in Appendix II, page 58. It follows closely the method used by Love* 
in his work on the theory of elasticity, to which those who wish to in- 
vestigate the subject further are referred. 

Six theories have been advanced to cover the problems of the 
strength of material under combined stress. Two of them are empirical, 
one is developed from a molecular hypothesis, one from the mathematical 
theory of elasticity, and two from static relations of stresses. Three of 
these theories have found considerable favor and are given first. 

6. The Maximum Strain Theory. — In the mathematical theory 
of elasticity, after the relations between stress and strain are established 
for simple stress, three equations of the following types are derived : 

Ee^ = (j^—— (o-o + o-g), 

where o-j, o-,, and o-g are the three stresses at right angles to each other, 
E is the modulus of elasticity assumed constant in all directions, c^ is 

the unit-strain in the direction of a^, and — is Poisson's raticf Stresses 

m 



*The Mathematical Theory of Elasticity, A. E. H. Love. 

tA stress in any direction produces strain in that direction and also strain at right 
angles to that direction. The numerical ratio between the unit-strain at right angles to 
the direction of the force and the unit-strain in the direction of the force is called 
Poisson's ratio. 



BECKER STEEL UXDER BIAXIAL LOADING 9 

are considered positive if tension, and negative if compression. Ee^ is 
called by various writers the reduced stress, the true stress, or the ideal 
stress, but as the term stress is generally used by engineers to mean an 
internal resisting force which holds external forces in equilibrium it 
seems best to refer to it merely as Ee. Writing two equations similar 
to the above for Ee.y and Ee.^^, the three equations for the reduced stress 
are obtained. The maximum strain theory takes these three equations 
and assumes that whatever the combination of stresses, the material will 
fail when the maximum strain (which will be in the direction of the 
greatest stress) reaches a value equal in magnitude to that at the yield- 
point stress in simple tension or compression. Ee at the yield-point 
stress for any combination of stresses must be the same, provided the 
yield-point stress is the same for tension as for compression. For 
ductile materials, E is usually assumed to be constant and it follows 
that € must be the same when the yield-point stress of the material is 
reached, no matter what combination of stresses is used. But for a 
brittle material, where E varies, the strain e must vary in an inverse 
ratio; that is, the product remains constant. 

The maximum strain theory, or St. Venant's theory as it is some- 
times called, holds that when a material is subjected to two or three 
stresses at right angles to each other, its strength is increased if the 
stresses are of like sign and that its strength is diminished if the stresses 
are opposite in sign. Thus two tensions or two compressions will pro- 
duce an increase in the elastic strength of the material, whereas a tension 
combined with a compression produces a reduction in strength. For a 
stress ratio of one to one, both stresses tension, the material will be 
increased in strength 43 per cent if Poisson's ratio is 0.3, while if one 
stress is tension and the other compression, it will be weakened 23 per 
cent for the same stress ratio. 

If in the equation for reduced stress given above, o-g and 0-3 are zero, 
the case is that of a bar in simple tension (compression is expressed as 
negative tension) and dividing both sides of the equation by e^, the result 
is the equation of the modulus of elasticity. 

For combined stress according to this theory, then, the strain ac- 
companying a given stress is changed by the addition of another stress 
at right angles to the first. It is increased if the stresses have unlike 
signs and diminished if they have like signs. Also, the strain c is the 
measure of Ee (the reduced stress) and the material will not reach the 
yield point until the strain e reaches the value corresponding to the 
strain obtained in simple tension at the yield point. It should be 



10 ILLINOIS EXGINEERIXG EXPERIMENT STATION 

emphasized that all elastic theory holds only within the elastic limit, or 
more correctly within the limit of proportionality, where E remaina^ 
constant for an individual stress-strain diagram. But the slight varia- 
tion up to the yield point, even though the value of E does change- 
slightly, does not invalidate the theory, and the yield point is commonly 
taken as the limit of the discussion. 

The maximum strain theory is based upon the mathematical theory 
of elasticity. Temperature effect is neglected and Hooke's law is assumed 
to hold rigidly. Herein lies its weakness, for the maximum strain theory,- 
like the mathematical theory of elasticity, is dependent upon the accuracy 
of the relation assumed between stresses and strains. It has been shown* 
that there is a cooling of a bar of metal as the stress is increased up to 
the yield-point stress and it is also well known that Hooke's law is only 
an approximation.^ A very good approximation it is, to be sure, for 
engineering purposes, but lack of isotropy in the materials, cold working 
and similar causes tend to change conditions, so that a slight deviation 
from Hooke's law may be observed considerably before the yield-point 
stress is reached. While the maximum strain theory has a good founda- 
tion, it must not be expected that the measured strains upon a body 
known to be not wholly isotropic, will conform exactly to this theory of 
stiffness. 

The question of strength is quite different, for there is no assurance 
that the strains are the true measures of strength. Eeasonable as the 
assumption may be, it is an assumption whose correctness must be 
demonstrated by experiment. 

7. The Maximum Stress Theory. — The maximum stress theory, 
or Rankine's theory as it is sometimes called, virtually assumes that what- 
ever the ratio of the stresses in the two directions and whether they are 
of like or opposite sign, the material will reach the yield point when, and 
only when, one of the stresses reaches the value corresponding to the 
yield-point stress in simple tension or in compression, as the case may be. 
It takes no account of Poisson's ratio as affecting strength and assumes 
that a material is neither weakened nor strengthened by the addition of 
a second stress at right angles to the first. If, then, this theory holds, 
the material should reach its yield point when the greater stress reaches 
the yield point stress for uni-directional loading. 

8. The Maximum Shear Theory. — In the preceding theories failure 



*C. A. p. Turner, Trans. Am. Soc. C. E., 1902. Lawson and Capp, Inter. Assn. Test. 
Mat., 1912. Ew. Rasch, Inter. Assn. Test. Mat., 1909. 
tHedrick, Engineering News, Sept. 16, 1915. 



BECKER — STEEL UNDER BIAXIAL LOADING 11 

by yielding is considered to take place in tension or compression, Avhereas 
the maximum shear theor}', or Guest's law as it is sometimes called, 
holds that all failures are failures by yielding due to shear when the 
shearing unit-stress reaches the shearing yield-point stress. Therefore, 
if loads are gradually applied to two specimens developing simple stress 
in one and combined stress in the other but so as to keep the shearing 
stresses the same in each specimen, the yielding failure in the two cases 
will be identical. 

The basic principle of the maximum shear theory, that the failure 
in combined stress is the result of the shearing stress reaching the shear- 
ing yield-point stress, when carried to its logical conclusion demands 
that when two of the principal stresses are zero the failure is still due 
to shear. A steel bar subjected to axial tension only must therefore fail 
in shear. The maximum shear in this case occurs on a 45° plane and its 
intensity is one-half the tensile unit-stress. If the yielding due to shear- 
ing stress occurs at the same time as yielding due to tensile stress the 
yield point unit-stress of the material in shear must be just one-half 
that in tension, but if the shearing yield-point stress is reached first — as 
this theory maintains — then the ratio is somewhat less than one-half. 

If the stresses which are combined are a compression and a tension, 
the resulting maximum shearing unit-stress is one-half the sum of the 
tensile and compressive unit-stresses. When the tensile and compressive 
stresses are equal, the intensity of the shearing stress is equal to the 
intensity of the tensile or compressive stresses and failure will take place 
by shear unless the shearing yield-point stress is equal to or greater than 
that of either tension or compression. It seems entirely possible, then, 
that failure may be caused under certain conditions by shear and that 
in other cases its intensity may be insufficient to cause yielding, the 
tensile or the compressive yield-point stress being reached first. 

Considering compression as negative tension, there are two kinds of 
elementary stress treated in mechanics — tension and shear. They aie 
quite distinct and have different accompanying phenomena. While a 
definite relationship may be established between the shearing and tensile 
stresses, the material may fail either in tension or in shear. This is 
suggested by the fact that mild steel in torsion gives a square break, a 
shearing failure, but cast iron tested in torsion breaks along a helicoid, 
failing in tension because the material is weaker in tension than in shear. 

This duality of conditions while not entirely overlooked, has been 
advanced heretofore solely to form two distinct theories of failure, but 
these have not been connected. The possibility that both shear and 



12 ILLINOIS ENGINEERIXG EXPERIMENT STATION 

tension may govern, each within certain limits, has apparently not been 
mentioned in the publications and discussions on this subject. Mallock* 
has stated a dual law which is quite different from that discussed above. 
He proposes a volume extension limit and a shear limit, each dependent 
upon the other, and assumes that the material will fail when the limit 
of either is reached. This is quite distinct from the simple stresses as 
controlling factors in the failure of the material, but it recognizes the 
possibility of dual control. 

The usual stress derivation for combined stress given in textbook 
is based upon the static equilibrium of forces and an application is made 
to a circular shaft in combined bending and torsion. A solution is given 
for the maximum normal stress and shearing stress on oblique planes, 
and safe working stresses are assigned. The assignment of working 
stresses in shear and tension fixes an arbitrary ratio of shear to tension, 
and the larger of the "h\^o shaft diameters determined by the two formulas 
is to be taken. 

9. The Internal Friction Theory. — A short cylinder of brittle 
material when tested in compression fractures by shearing along a 
diagonal plane which, if failure be due to shear, should make an angle 
of 45° with the axis, since this is the plane of greatest shearing intensity. 
But the angles observed in experiments differ from 45°. In the attempt 
to explain this variation the theory of internal friction has resulted. 
When two particles under stress tend to slide over each other, a condi- 
tion is set up similar to that of ordinary sliding friction. On the sup- 
position that this resistance is similar to sliding friction, one of the laws 
governing the latter is applied; namely, that the coefficient of internal 
friction is independent of the load or stress. Therefore, slipping will 
occur along the surface of the plane inclined at an angle p with the 

axis of the specimen such that ^ = 45° — — for compression and 

/? = 45° +-■ for tension. 4> is the angle of friction and tan <f> = jx, 

tho coefficient of friction. If the limiting friction per unit of surface 
is the same for tension and for compression, then the normal stress on 
the surface of slipping, at the instant when yielding begins, must be 

the same in each case, since this is - times the limiting friction. 

It has been said that the chief difference between the internal fric- 
tion theory and the maximum shear theory is that the former is based 



•Proc. Royal Society of London, 1909. 



BECKER STEEL UNDER BIAXIAL LOADING 13 

upon a maximum resistance to sliding, while the latter is based upon a 
maximum shearing stress. If the angle of friction is zero, the internal 
friction theory becomes the maximum shear theory. 

10. Mohrs Theory.* 

Let I'i = the shearing yield-point stress. 

Let Z'j = the stress in compression and in tension (equal) which to- 
gether produce a shearing stress equal to the shearing yield-point 
stress, Ic^. 

Let Ji\ = the tensile yield-point stress. 

Let A% = the compressive yield-point stress. 

Mohr derives the formulas : 

k, h, -, , _ 1 



^3 = ^. _^^ and ^-4 = 2^ ^^i ^'2 

The usual theory developed from the static relation of stresses gives 
for two equal stresses of unlike sign the following relation for the stress 
intensities: 

Shearing stress = ^ (tensile stress + compressive stress) which is 
the same as Mohr^s theory when the tensile and compressive yield-point 
stresses are equal. Mohr^s theory is an attempt to modify the shearing 
yield-point stress according to the tensile and compressive yield-point 
stresses. When these are equal this theory presents nothing new, for it 
then coincides with the maximum shear theory. If the yield-point stresses 
are different, Mohr's theory brings in a new relation regarding the shear 
failure in combined stress. It is virtually an acceptance of the maximum 
shear theory with the definition of the value of that shear at the yield- 
point. 

11. Wchage's Theory.j — This theory is based upon a few experi- 
ments on cross-shaped pieces of paper submitted to tension in two direc- 
tions at right angles to each other. If the material has a different yield- 
point stress in the two directions, the following elliptic relation is given 
as an empirical deduction : 

J\ and To are the yield-point or the ultimate stresses in the two 
directions (as, for instance, with and across the direction of rolling), 
and t^ and fo are the applied stresses in the corresponding directions. 
When 1\ — To, this elliptic relation becomes a circular one. 

This theory assumes that the material is ivealcened by the applica- 

*Zeitschrift des Vereines Deutcher Ingenieure. 1900. 
tZeitschrift des Vereines Deutcher Ingenieure, 1905. 



14 



ILLINOIS ENGINEERING EXPERIMENT STATION 



tion of two tensions for the reason that such stresses tend to lessen the 
cohesion between the fibers. The assertion is also made that a com- 
pression combined with a tension should strengthen the material by 
increasing this cohesion, although no formula is proposed. 

12. Graphical Presentation of Three Theories. — A graphical pre- 
sentation frequently serves to give a better idea of the working of a 
theory or formula and for this reason the three most important theories 



Compression 



Tension 




Fig, 2. Graphical Representation of Stresses According to the Maximum 

Strain Theory. 



are represented in Fig. 2, 3, and 4, for the four combinations of simple 
tension and compression. To make the presentation more general, differ- 
ent yield-point stresses in compression and in tension have been assumed 
where this is possible. 

Maximum Strain Theory. Let OA (Fig. 2) and OB represent the 
yield-point stress in simple tension and OC and OD that in compression. 
A tensile stress equal to OE would require a tensile stress equal to OF 
at right angles to cause yielding. For two equal tensile stresses the 
condition of yielding would not be reached until each stress attained 
the value OG, equal to OH. The increase in strength is OG — OB. 

For a compression combined with an equal tension, yielding would 
occur when each stress attained the value IST, equal to M. The other 
two quadrants are similar, two compressions producing the same relative 



BECKER — STEEL UNDER BIAXIAL LOADING 



15 



effect as two tensions, and a tension and compression producing a 
corresponding effect to a compression and a tension. 

Maximum Stress Theory. Yielding takes place in tension or in 
compression and since the stress in one direction is not affected by a 
second stress at right angles to the first the diagram will be a square. 
The center of the square, however, will not be the origin of co-ordinates 
since the tensile and compressive yield-points will in general be different. 
If a tensile stress OB or a compressive stress OD, Fig. 3, equal to the 
yield-point stress, is applied in one direction, any stress, OE less than 
the yield-point stress in tension, may be applied at right angles without 
causing further yielding. In other words a second stress acting at right 



Q 


1 


/J 


/r 


P 
Compression 




,/f 


p 







g Tens/or? 


D 






1 


L 


^ 


c 





Fig. 3. 



Graphical Representation of Stresses According to the Maximum 
Stress Theory. 



angles to the first yield-point stress does not change the yield-point 
stress of the material. 

Maximum Shear Theory. The first and third quadrants (Fig. 4) 
correspond to the maximum stress theory. This follows from the fact 
that the shearing stress equals one-half the difference between the great- 
est and the least of the three principal stresses. For biaxial loading one 
of the three principal stresses is zero and in the first and third quadrants 
the other two are of like sign, hence the shearing stress will be one-half 
the greatest stress. But the limiting shearing stress must be constant. 



16 



ILLINOIS EXGINEERING EXPERIMENT STATION 



therefore the greatest limiting principal stress must be constant and 
for like stresses (first and third quadrants) the diagram corresponds 
to the maximum stress theory. For a combination of tension and com- 
pression (second and fourth quadrants) the lines CB and AD are 
inclined at an angle of 45°, because the tensile stress plus the compressive 
stress is a constant and is equal to twice the shearing stress. 

t -\- c = constant. 
By setting t and c each equal to zero in turn, it is seen that t must 
€qual c, and this theory demands an equal yield-point stress for tension 




Fig. 4. Graphical Representation of Stresses According to the Maximum 

Shear Theory. 

and compression. Two equal stresses of unlike sign will then cause 
yielding of the material when each stress equals ON or OM. 

III. Experimental Work. 

13. Form of Specimen. — The selection of the type of specimen to 
be used in the experimental work was a problem of considerable difficulty. 
Specimens subjected to direct tension or compression in two directions 
were not considered because of complications produced by the method of 
application of the load. A cube subjected to compression in two direc- 
tions could easily have been set up, but the friction between the bearing 
blocks and surfaces of the cube introduces inequalities and resistance 
to the change in cross section which could easily vitiate the rosults.* 

•See Zeitschrift des Vereins Deutscher Ingenieure, 1900, p. 1530. 



BECKER — STEEL UNDErx BIAXIAL LOADIXG 17 

A large number of short square steel bars, closely spaced to form in effect 
a bearing block, were considered not to obviate this difficulty sufficiently. 
Similarly, a tension specimen held at the four edges would not be 
practicable. Direct stress application seemed out of the question, and 
recourse was first had to bending to produce stresses in two directions 
at right angles to each other. 

The first biaxial stress experiments in this series of tests were made 
upon flat cross-shaped specimens subjected to cross bending to produce 
two compressions or two tensions at right angles to each other. The 
stress distribution was so far from regular that no safe comparisons 
could be made. Such difficulties were encountered that this form of 
test specimen was discarded. 

After a preliminary test, thin tubes were adopted as the form of 
test specimen. They proved satisfactory on account of the certainty 
with which biaxial stress of known magnitude could be applied by means 
of an axial load in a testing machine and internal hydrostatic pressure 
producing a circumferential tension. This method gives two well defined 
principal stresses at right angles to each other, the stress in the third 
direction being small since it varies from the intensity of the hydro- 
static pressure on the inside to zero on the outside. It is much easier 
to cover the total range of stress ratios by the use of hydrostatic pres- 
sure and axial tension or compression in the tubes, than to use torque 
and axial load on solid bars. The latter method is inferior to the tube 
tests since only a small portion of the material is carried to the yield- 
point stress. The experiments are more successful when as much of the 
specimen as possible is uniformly stressed, and the best condition is 
that wherein the entire specimen is uniformly stressed. This is true both 
on account of the pronounced yield-point effect and on account of the 
smallness of the strains to be measured. The thinness of the wall and 
the relatively large tube diameter made the stresses practically uniform 
throughout the tube. It may be expected that the stress-strain diagrams 
will show a much sharper break than for solid bar specimens and the 
yield point is more positively determined. There are no greater eccen- 
tricities of application of load when using the tube than when working 
with a solid bar, and on account of the greater diameter of the tube, this 
eccentricity is relatively less important. 

Strains were measured by means of a Berry strain gage, using a 
2-in. gage length in the cross bending tests and a 4-in. gage length 
in the tube tests. The accuracy and reliability of an instrument of 



18 



ILLINOIS ENGINEERING EXPERIMENT STATION 



this type has been demonstrated repeatedly and reference is made 
to the tests by A. N. Talbot and W. A. Slater on reinforced con- 
crete buildings, as given in Bulletin No. 64 of the Engineering Experi- 
ment Station of the University of Illinois, to show what results may be 




Fig. 5. View Showing Cross-Bending Test Specimen Under Load. 



achieved with such an instrument. A discussion of the strain gage and 
its use is given in a paper by Slater and Moore in Vol. XIII of the 
Proceedings of the American Society for Testing Materials. 

The use of the strain gage marks a decided advance in the measure- 
ment of strains. With this instrument it was possible in these tests 
to take twenty-eight readings on as many gage lines for each increment 



BECKER — STEEL UNDER BIAXIAL LOADING 19 

of load, whereas other investigators have been able to take four at the 
most and often only two. The advantage of a portable instrument 
over an attached one is very great and the rapidity of operation and 
freedom from danger of jarring the instrument as well as the ability 
to read overlapping gage lines, as was done in these tests, marks a 
decided step in advance. 

14, Cross-lending Tests. — The set-up for the bending tests is 
shown in Fig. 5. Two specimens were prepared from l^-in. soft steel 
plate of the shape shown in the figure. Tension specimens were pre- 
pared from the portions cut away. In order to have the upper surface 
unobstructed for the use of the strain gage, the beam was loaded as an 
overhung beam with four equal loads placed symmetrically one on each 
projection of the cross-shaped specimen. The center part of the cross 
was thus subjected on the top to two tensions at right angles to each 
other. 

Load was applied by placing known weights on the yokes at the 
ends of the arms of the specimen, thus giving a definite bending moment. 
The strains were mtasured over 2-in. gage lines with a Berry strain gage. 
Instead of a uniform stress over the center portion of the test piece, the 
readings showed a considerable variation. The effect of the sharp re- 
entrant angles at the corners in changing the lines of stress must have 
been considerable, for the yield point was reached first at the corners. 
The lines of yielding spread inward along a line making an angle of 
approximately 45° with the center lines. As the load was increased 
these lines divided, curving toward the adjacent corners, gradually 
changing direction and becoming parallel to the lines of symmetry of 
the specimen shortly before the lines from adjacent corners joined. New 
lines formed beside the first ones and others appeared outside the center 
of the cross. The latter were straight and parallel to the support. The 
lines are clearly shown in Fig. 6, which is from a photograph of the 
compression side of the first specimen tested. The lines marking the 
square from corner to corner and the center lines were used to lay out 
the specimen and must not be confused with the lines of yielding. The 
specimen, considered as a beam, widens abruptly for the center four 
inches, but the effect of this increased width in carrying stress was slight. 
The places of greatest stress were near each corner and to measure the 
maximum strain would have required a very short gage line. This 
stress condition is due to the form of the specimen rather than to com- 
bined stress. 

15. Tube Specimens. — Specimens made from 6-in. tubes with 



20 



ILLINOIS EXGIXEERIXG EXPERIMENT STATION 



1/4-iii. walls were used. Four lengths of seamless drawn tubing were 
bought in the open market and made into test specimens. A series num- 
ber was given to the specimens cut from a length of tubing and each 
specimen was numbered individually. The number of the test specimens 




Fig. 6. View Showing Compression Side of Cross-Bending Test Specimen 

After Test, 



in each series cut from each length of tubing and the character of the 
stresses applied are as follows: 



Character of Combined Stress 
Tension with tension 
Tension with tension 
Compression with tension 
Tension with tension 





Number of 


Specimen 


Series 


Specimens 


Number 


1 


5 


1-2-3-4-5 


2 


4 


6-7-8-9 


3 


6 


1-2-3-4-5 


4 


5 


8-9-10 



BECKER STEEL UXDEK BIAXIAL LOADING 



21 



Tube No. C) of Series 3 and tube No. 7 of Series 4 were tested in 
torsion only. There was a marked difference in the physical properties 
of the material of the four lengths of tubing. This is shown by the 
stress-strain diagrams of the tensile tests made on specimens cut from 
the tubes. The yield point stress varied from 21,500 lb. per sq. in. to 




'/i s c /? 

Fig. 7. Dimensions of Tube Test Specimen. 



je f/7af5.//h. 



50,000 lb. per sq. in., Series 1, 2, 3, and 4 having yield-point stresses of 
42,500, 21,500, 24,000, and 50,000 lb. per sq. in. respectively. The 
tubes were not annealed, but the first three series gave very uniform 
results for all gage lines, and showed a decided change at the yield point. 
The specimens of Series 4 showed a much greater variation. The be- 
havior was that of hard, brittle steel of quite irregular composition. 




Arrangement of Gage Holes on Tubes. 



There was little reduction of area and the rupture was sharp and 
sudden, both in the tension specimens and in the one tube that broke 
during testing. The stress-strain diagram for the Series 4 show only 
qualitative results. These tubes were not suited for a test of this charac- 
ter, the inner and outer circumferences of the tube before machining 
were not concentric circles, and some gage lines gave diagrams that 



22 



ILLINOIS EXGIXEERING EXPERIMENT STATION 



curved throughout, similar to the diagrams of drawn wire. There was 
no well-defined yield point. As the stress-strain diagrams did not give 
positive results, no use will be made of this series. 

16. Preparation of the Tubes.~The test specimens were first 




Fig. 9. Location of Gage Lines on Developed Outer Surface of a Tube. 

bored out for the entire length on a horizontal boring mill and then 
turned to the dimensions shown in Fig 7". Each tube was threaded on 
the two ends with a taper thread of twelve threads per inch over a length 



BECKEK — STEEL UXDEIJ BIAXIAL LOADING 23 

of three inches. The tube was left full thickness for about an inch 
beyond the threads to furnish a bearing for packing. The remainder of 
the tube was turned to an approximate thickness of 3/32 in. except 
for four bands of 1/4-in. width spaced four inches apart along the tube. 
The greater part of these bands Avere afterwards milled off leaving four 
projections on each band for the gage holes. The tube was thus spanned 
with four circumferential gage lines each four inches long. The axial 
gage lines used one of the two holes so that the projections could be 
reduced to the smallest possible size. This gave four rows of three 
axial gage lines each, twelve in all, and four bands of four circum^ 
ferential gage lines, sixteen in all, making it necessary to take twenty- 
eight readings, exclusive of the standard bar and check readings for 
each increment of load. The standard bar readings are necessary in 
tests with the strain gage to detect variations in the instrument due to 
temperature or jarring of the points. 

Fig. 9 shows the position of the gage lines on a developed surface of 
a tube specimen. The circumferential bands were lettered A, B, C, and 
D; the axial lines were numbered 1, 2, 3, and 4. Thus an axial gage 
line would take two holes in the same axial line, but in two consecutive 
circumferential bands. It w^ould, consequently, be called by the letters 
of the bands, in order, and by the number of the axial line. Thus AB 3 
would be an axial gage line spanning the distance between the circum- 
ferential bands A and B and lying along the axial line 3. As soon as 
the tube was machined the numbering was fixed and the projections on 
the A band marked with small prick punch marks to identify the axial 
lines. In this way the readings for the thickness of the tube walls could 
be correlated with the strain gage readings. The gage holes were drilled 
by hand using a Xo. 54 drill. They were not reamed. 

The boring of the tube caused a slight change of shape of the 
cross section due to the removal of the inner skin of metal, and after the 
outside was turned the thickness was uniformly varying, usually having 
two points of maximum thickness diametrically opposite, and at 90° 
from these, two points of minimum thickness. This renders the tube 
slightly elliptical (but not over 0.02 in. in 5.50 in.) and of varying thick- 
ness. While the variation in thickness was as high as 15 per cent in 
some cases, it apparently did not affect the averages of the readings, 
although the individual circumferential curves show the effect of this 
variation and the effect of the water pressure in making the tube more 
nearly cylindrical. 



21 



ILLIXOIS EXGIXEERIXG EXPERIMENT STATION 



17. Deiermitmtion of the Thickness of Tube Walls. — The principle 
of the apparatus adopted for measuring the thickness of the tube walls is 
that a micrometer caliper with a very deep throat. Fig. 10 shows the 
apparatus with the tube in position for a zero reading. A 4l^ by 21/2 by 
7/16-in. T-bar was clamped at one end to a support and a stiff wooden 
bar was bolted to it. At one end of the wooden bar an Ames Dial read- 




FiG. 10. View of Apparatus Used ix AIeasuring Thickness of Tube Walls. 



ing to thousandths of an inch was fastened so that the plunger rested on 
a steel ball {a Fig. 10) embedded in the stem of the T-bar. To determine 
the thickness of the tube wall the plunger of the dial was raised, the tube 
was slipped over the T-bar and rested on the steel ball. Two other steel 
balls ( h and c Fig. 10) were embedded in the stem of the T-bar, one 
on each side of the ball under the plunger at such a distance from it that 
the tube always swung free on the center ball and one of the others. The 
ball under the plunger was slightly higher than either of the others to 
insure a bearing on it at all times. When the plunger of the dial was in 
contact with the tube, the thickness of the tube was the difference between 
the reading then taken and the zero reading. Zero readings were ob- 
tained by suspending the tube in two fine wire slings in such a manner 



BECKER — STEEL UNDER BIAXIAL LOADIXG 25 

that its weight came on the T-bar in the same way as when tl\e tube 
swung on the steel balls. With the plunger of the dial in contact with 
the steel ball, the initial or zero reading was taken for every position of 
the tube along an axial line. As the T-bar was a cantilever with two 
point loading, this gave slightly different zero readings for the various 
positions of the tube, but any error arising on account of the deflection 
of the apparatus was removed. With the tube in a given position and 
with the plunger on the ball, a reading was taken after a traverse of two 
axial lines. These readings were taken to detect any possible change in 
the apparatus and are not zero readings. They correspond to the 
standard bar readings when using a strain gage. A set of check readings 
was taken and the average of the two readings was used. Readings were 
taken to tenths of a division (ten-thousandths of an inch) and tube thick- 
nesses are given in thousandths of an inch. 

It is thought that this method of measurement is accurate and the 
check results obtained with a micrometer after the tube had been cut, 
have boine out this conclusion. The tube must be of relatively large 
diameter to apply this method, but with 6-in. tubes no difficulty was 
experienced. 

18. Method of Testing. — Two steel castings were designed to fit 
over the ends of a tube. The stresses carried by these heads were com- 
paratively low, for the maximum load was but 167,000 lb., and the 
material was about % in. thick at the thinnest part. The castings were 
machined all over and threaded internally, at one end to receive the tube 
and at the other to receive a 4-in. bar which served to apply the tension. 
The two threaded portions were separated by about an inch of metal 
which served to retain the water under pressure in the tube. These 
castings are shown in Fig. 11. 

To withstand the water pressure, two layers of %-in. hydraulic 
packing were used in an ordinary four-screw stuffing box. The heads 
were recessed to receive the packing and the gland, while the tube walls 
were left nearly full thickness for about an inch beyond the threads to 
furnish a firm bearing for the packing. After the packing was ad- 
justed to position there were no perceptible leaks although pressures 
up to 1,800 lb. per sq. in. were used. Fig. 11 shows the general arrange- 
ment of the apparatus for the tension tests. 

All the tests except the torsion tests were made in the 600,000-lb. 
Riehle machine of the Laboratory of Applied Mechanics of the Uni- 
versity of Illinois. By using spherical seats with careful centering of 
the specimens in the machine, the eccentricity of loading was reduced 



26 



ILLIXOIS EXGIXEERING EXPERIMEXT STATIOX 




Fig. 11. View Showing Arrangement of Apparatus for Tension Test. 



BECKER STEEL UNDER BIAXIAL LOADING 27 

to a minimum and the bending stresses were low as shown by the uni- 
formity of the individual stress-strain diagrams. In the tension tests 
the nuts of the 4-in. bars bore directly against the spherical seats, the 
lower one being inverted. For the compression tests the bars were re- 
moved and the tube heads bore directly against the spherical seats and the 
upper spherical seat was inverted. The length of thread on the specimens 
tended to give a good distribution of load and the distance of the first gage 
hole from the end of the thin part of the wall (6I/2 in.) together with the 
thinness of the wall itself, were sufficient to insure a high degree of uni- 
formity of stress. Holes were drilled into each head to connect into the 
interior of the tube ; the hole in the lower head was for connection to the 
pump and the hole in the upper head was for the purpose of filling the tube 
with water. Each hole w^as tapped with a %-in. pipe tap. 

The torsion tests were made in a 230,000-lb. in. Olsen Torsion 
Machine. The heads were screwed on the specimens as in the other tests, 
and short steel bars threaded on one end transmitted the torque from the 
machine to the steel heads. Two fixed wooden clamps with 40-in. arms, 
one of which carried a pointer and the other a scale, were used to meas- 
ure the angle of torsion over a known gage length. 

19. Character and Sequence of the Tests. — Three tests in each of 
Series 1 and 2 were first made. These were the tests in direct tension, 
the tests with the ratio of tensile stresses equal to 0.475 and with this 
ratio equal to 0.92. The results of these tests were worked up before the 
remainder of the tests in the series were made, so that the other ratios 
could be chosen to the best advantage. As the difference in the strength 
of the steel in the direction of drawing and across it would complicate 
the problem, and as it was not intended to raise the question of the 
variation in strength in different directions throughout the specimen, 
the highest ratio of circumferential stress to axial stress used was made 
less than 1.0, being 0.92. 

The average area of the inner cross section of the tubes was about 
24.50 sq. in., and the axial load due to the water pressure was 2,450 lb. 
per 100 lb. per sq. in. water pressure. To produce a ratio of circum- 
ferential tension to axial tension equal to 0.50 required a machine load of 
4 X 2,450 — 1 X 2,450 = 7,350 lb. per 100 lb. per sq. in. water pressure, 
since the water pressure acts with the machine load. For axial com- 
pression combined with circumferential tension, the two quantities 
would be added instead of subtracted, since the water pressure tends to 
reduce the machine load. Dividing the net axial load (9,800 lb. per 



28 



ILLINOIS ENGINEERING EXPERI]SLENT STATION 



100 lb. per sq. in. water pressure) by the cross-sectional area of the 
tube gives the unit axial stress. A slight error is introduced by using 
the inside diameter of the tube rather than the mean diameter, for in 
order that the circumferential tension shall be exactly twice the axial 
tension when the water pressure alone is acting, the mean diameter must 
be used to compute the axial tension. This error is about II/2 per cent, 
which repiresents the variation of the circumferential tensions from the 
mean. The stress ratios for Series 3 and 4 were planned complete and 
carried out as planned. The stress ratios used in the four series are 
given in Table 1. 

The strain gage used was a 4-in. Berry strain gage made for the 
Joint CommittcQ on Stresses in Railroad Track and loaned by that Com- 
mittee. It has invar steel sides and shows a negligible correction for 
temperature. Two standard bars were used to detect any variation of 
the instrument due to jarring or striking the fixed point. All data have 
been corrected for variation in the standard bar readings. To avoid 
variation due to change of temperature of the tubes, they were usually 



Table 1. 
Outline of Test Specimens and Tests. 



Series 

No. 


Tube 
No. 


Ratio of Circumfer- 
ential to Axial Stress 


Stress Combination 




5 


0.00 


Axial tension only 




1 


0.24 


Tension with tension 


1 


2 


0.475 


a a (I 




4 


0.69 


(I a a 




3 


0.92 


ii a « 




9 


0.00 


Axial tension only- 




7 


0.475 


Tension with tension 


2 


8 


0.92 


« a a 




6 


0.92 


" 




4 


0.00 


Axial compression only 




2 


0.20 


Compression with tension 


n 


5 


0.30 


11 <( « 


3 


3 


0.60 


« <( « 




1 


0.90 


" " " 




6 


1.00 


Torsion only 




9 


0.00 


Axial tension only 




10 


0.30 


Tension with tension 


4 


8 


0.50 


" " " 




11 


0.80 


"■ " " 




7 


1.00 


Torsion only 



BECKER — STEEL UNDER BIAXIAL LOADING 29 

filled with water in the evening and by the time the test began the next 
day the tube and w^ater were at a temperature that scarcely changed 
during the entire test. 

20. Test Operations. — The initial load in all cases was small, pro- 
ducing an average axial unit-stress of approximately 4,000 lb. per sq. 
in. Thie load was applied after the specimen had been carefully centered 
and the spherical seats tried. A load sheet was prepared for each test 
which gave the required machine loads, the approximate yield point, 
tlie water pressure, and unit-stress. When the load was increased the 
water pressure was increased first and then the machine load. 

The record of a test was a combination of the ordinary record and 
a graphical one. Co-ordinate paper was used and was divided into a 
series of rectangles, one for each standard bar and gage line. Along one 
side of this rectangle the instrument reading was noted and this reading 
was then plotted against the machine load. In this way the progress 
of the test was very evident and any doubtful reading was checked. 
When the nature of the curve is well known, it is advisable to see that 
the results for any gage line that do not show some systematic sequence 
of plotted points are checked to insure their accuracy. If this is not 
done false breaks may sometimes be obtained in the curve. If the error 
is experimental, the check reading will correct it, and if the stress sud- 
denly departs from the straight line law, the check reading will be a 
repetition of the first reading and will give greater confidence in the 
result. Though but few errors were discovered and corrected, the result 
justifies the method employed. Whenever there are variations from 
the straight line in the stress-strain diagram, these are indications of a 
change in the rate of taking stress. As the load changes, the distribu- 
tion of stress over a given cross section often changes, so that at one 
point there may be a rapid increase in the elongations for one increment 
of load, while in an adjoining gage line the change is slight. The next 
load increment may bring about a complete reversal of the conditions 
shown by the previous instrument readings. 

Whatever variation occurs in one gage line, usually it is reflected in 
one or more of the others, so that the average takes out all these peculiar- 
ities. This is especially true of the circumferential readings. 

It will be seen that the circumferential gage line readings give the 
correct unit-strain, the chord length being used and not the arc length. 
Circumferential readings are subject to the tendency of the tube to 
become truly cylindrical under water pressure. For low water pressures 



TABLE 2. 
Data of Tubes. 



Tube No. 


Inside 

Diameter 

Inches 


Location 


Tube Walls. 

Average 

Thickness, 

Inches 


Sectional 

Area, 

Sq. In. 






Series ]. 




r 


1 


5.564 


AB 
BC 
CD 


0.089 
0.088 
0.086 


1.590 
1.570 
1.527 


2 


5.558 


AB 
BC 
CD 


0.087 
0.087 
0.088 


1.543 
1.543 
1.570 


3 


5.561 


AB 
BC 
CD 


0.087 
0.087 
0.087 


1.543 
1.543 
1.543 


4 


5.563 


AB 
BC 
CD 


0.085 
0.084 
0.083 


1.508 
1.481 
1.472 


5 


5.554 


AB 
BC 
CD 

Series 2. 


0.091 
0.092 
0.092 


1.623 
1.641 
1.641 


6 


5.579 


AB 
BC 
CD 


0.083 
0.083 
0.080 


1.467 
1.467 
1.422 


7 


5.588 


AB 
BC 
CD 


0.091 
0.091 
0.091 


1.623 
1.623 
1.623 


8 


5.560 


AB 
BC 
CD 

Series 3. 


0.094 
0.094 
0.094 


1.678 
1.678 
1.678 


1 


5.573 


AB 
BC 
CD 


0.106 
0.106 
0.112 


1.891 
1.891 
2.001 


2 


5.561 


AB 
BC 
CD 


0.114 
0.111 
0.111 


2.040 
1.981 
1.981 


3 


5.566 


AB 
BC 
CD 


0.108 
0.107 
0.106 


1.934 
1.912 
1.889 


4 


5.581 


AB 
BC 
CD 


0.116 
0.114 
0.112 


2.076 
2.041 
2.004 


5 


5.622 


AB 
BC 
CD 


0.094 
0.094 
0.096 


1.688 
1.688 
1.724 


6 


5.634 


AB 

BC 
CD 


0.084 
0.084 
0.083 


1.509 
1.509 
1.490 



30 



BECKER — STEEL UNDER BIAXIAL LOADING - 31 

this was sufficient in some cases to change the stress from a tension to a 
compression or vice versa. 

21. Diagrams and Tables. — Stress-strain diagrams representing 
the general average results of the axial and circumferential gage lines 
are given in Fig. IT, 18, and 19, while sample diagram showing the 
average results at different sections of the tube for both the tension- 
tension and the compression-tension experiments are given in Fig. 13 
to 16. Diagrams of the experimental results of Series 1 and 2 are to be 
found in Fig. 22, and those of Series 3 in Fig. 23. A comparison of the 
theories of the strength of materials under combined stress is made in 
Fig. 24, while Fig. 26 and 27 illustrate some of the work of other investi- 
gators. An outline of the test specimens and tests and the principal data 
of the tubes are given in Tables 1 and 2. Table 3 is given as a sample 
of the data for a single tube, tube No. 4 of Series 1. These data have 
been reduced and corrected for standard bar readings. All the original 
and reduced data as well as the stress-strain diagrams are on file at the 
Laboratory of Applied Mechanics of the University of Illinois. 

IV. Discussion of Eesults. 

22. The Criterion of Strength. — There are three possible stress 
limits any one of which may be the criterion of the strength of material 
— limit of proportionality, yield point, and rupture or ultimate strength. 
It is recognized that there may be a sharp distinction between the laws 
governing ductile materials and the laws governing brittle materials, for 
such a distinction is observed in the stress-strain diagrams and in com- 
pression and torsion failures. Since this discussion is limited to ductile 
materials, conditions will be treated only as they apply to such materials. 

It would appear at first thought that the limit of proportionality 
would be the proper basis upon which to determine the relative strength 
of material. The mathematical theory of elasticity is based upon 
Hooke's law generalized, engineering practice bases its computations 
largely upon this same law, and several investigators have used the 
limit of proportionality (which they called the elastic limit) as their 
criterion, notably Hancock* and Turner.f 

The limit of proportionality, or p-limit, is defined as the stress at 
which the constancy of the ratio of stress to strain ceases; that is, the 
modulus of elasticity is a constant up to this stress. It is often stated 



'American Society for Testing Materials, 1905, '06, '07, '08. 
[Engineering, London, February 5, 1909. 



32 



ILLINOIS EXGINEERING EXPERIMENT STATION 



Table 3. 

Test Data of Axial Gage Lines Tube No. 4, Series 1. 

Eatio of Circumferential Tension to Axial Tension 0.69. 
Inside Diameter of Tube 5.563. 



AB Gage Lines 
Average Thickness of Tube .06^/n. Area of 3ect/on I.SOBsqJn 



'Manrr Pressure 
lb.per IKjJn. 


Axia/Looa 

due to 

W.Pressurt 


l^achim 


Total 


Ay. /I^ia/ 


Reading on Cagel/'/^e 


Differences 


Av. 


Av 


pound. 


pounds 


lbperi<fin 


/IB/ 


AS2 


AB3 


/le^- 


/iBi 


/IBZ 


AB3 


.^B^ 


Diff. 


E/ongar/o, 


Gag^FH^ 


Con^Tvi 


lOO 


1 00 


2-^30 


s/oo 


750O 


43SO 


80.I 


/as 


2B.0 


62. S 




















300 


300 


7290 


1^1 00 


2/400 


/4200 


77.0 


5.9 


Z..S 


58/ 


3./ 


.5.0 


6./ 


4-4 


4-7 


.00024 


soo 


SOO 


/2/SO 


zzoso 


3S200 


23300 


72.5 


00 


/7./ 


S2 7 


7.6 


/0.9 


/O.B 


s.e 


9 a 


00049 


700 


700 


17 OOO 


32/00 


^e/oo 


32500 


6(0/ 


94.0 


I/.9 


4^78 


/4:0 


/6.9 


/&./ 


74 7 


/5.4 


.00077 


300 


BOO 


z / soo 


4//00 


esooo 


■^1 700 


6/0 


369 


5.0 


4/9 


i9./ 


2 4.0 


23.0 


20.6 


2/7 


O0JO9 


3SO 


3SO 


23100 


4-34-O0 


eesoo 


■q-^OOO 


60.3 


84.1 


3.5 


4-03 


/9.e 


26a 


24.5 


2/6 


23.Z 


00/ /e 


rooo 


/OOO 


Z4300 


4seoo 


69900 


4e200 


59 


82.0 


/.e 


39 7 


2/./ 


28.9 


26 2 


223 


24.e 


00/ 2 4 


1 oso 


lOSO 


2S£00 


47900 


734-00 


^8 SOO 


^73 


790 


99.0 


379 


22/ 


3/9 


290 


246 


26.9 


00/35 


/ /oo 


/oeo 


2 a SOO 


SO/OO 


76600 


S0900 


5S2 


7e.4 


97.7 


361 


2^9 


34.5 


30 3 


26 4 


23.0 


. 00 /4S 


n^o 


/ /40 


27 70O 


5Z400 


eo/oo 


53/00 


53/ 


73.3 


953 


340 


27.0 


376 


32 7 


28.5 


3/5 


00/53 


/200 


1/90 


2 92O0 


6^500 


ezsoo 


SS300 


50.9 


690 


93.0 


32/ 


^S.Z 


4-/9 


35.0 


30.4- 


34-/ 


00/7/ 


IZSO 


/2-^0 


30/00 


jseeoo 


aesoo 


57eoo 


■^70 


e53 


ago 


2a.z 


33/ 


456 


40.0 


343 


383 


00/32 


/300 


IZ30 


31300 


ssioo 


90400 


eoooo 


30.9 


579 


7f-.7 


/&/ 


^9.2 


530' 


53 3 


4-64- 


50.5 


C0253 



BC Oage Lines 
Average T/^/ckr/e35 of Ta^e .08-f/n Area of 3ecf/on /.48/ sg/n. 



Water Pressure 
lb person 


Axia/laad 


Machine 


Tata/ 


A y Axial 


/?ead/ng on Gage l/ne 


Differences 


Ay 


Ay 


W Prepare 


pounds 


pounds 


Ibpersqin. 


BCi 


ec2 


BC3 


BC4- 


BCi 


BCZ 


BC3 


BC^ 


D/ff ■ 


Eior^>or) 


GagefTdf. 


Corrected 


/OO 


/OO 


2430 


5/00 


7500 


4-oeo 


92.1 


795 


35.0 


£3.5 




















soo 


300 


7Z90 


/4/00 


2/400 


/440O 


37.0 


74B 


29.9 


58.4 


5./ 


•*. 7 


5.1 


5.1 


50 


.000 25 


.500 


500 


/ 2/50 


23 0S0 


35 200 


23900 


ai.9 


690 


24.7 


S3.e 


/oz 


/ o.s 


/ 0.3 


3.3 


/OZ 


.0005/ 


700 


700 


/7000 


3Z/00 


43/00 


33/00 


76.0 


63.9 


/'S.I 


■4ao 


7 6/ 


/S6 


75.9 


15.5 


/58 


.000 79 


300 


SOO 


2/ SOO 


4/ /oo 


62000 


4 2 soo 


ea.9 


57-Z 


/SO 


42 9 


23.Z 


22.3 


22.0 


20.6 


22.0 


OO//0 


950 


3SO 


23/00 


43400 


6 6SOO 


44800 


&7Z 


5SO 


//.o 


■4/2 


24.3 


245 


240 


ZZ.-i 


239 


oa/20 


/OOO 


7000 


2 4300 


4^5600 


69900 


4 7200 


650 


53.5 


/03 


4-0 


27/ 


26.0 


247 


23 S 


25.4 


.00/27 


/ 050 


/ oso 


2S500 


47900 


734-0O 


4-3400 


(.2.S 


5/5 


a. 4 


385 


2 9.6 


ZB.O 


2(2.6 


250 


273 


00/37 


//oo 


/090 


26SOO 


50/00 


76600 


5/ 900 


599 


4 9.2 


6.9 


366 


32.Z 


30.3 


28/ 


Z69 


294- 


.00/47 


//50 


1/40 


27700 


52400 


eo/oo 


S4/00 


56 3 


4 7.4 


4. a 


343 


358 


32 1 


30.Z 


29Z 


3/8 


.00/53 


/200 


//SO 


29200 


54500 


83500 


S&'^O 


52 3 


44.8 


/,5 


30 3 


39a 


34.7 


335 


332 


353 


00/77 


/250 


/24-0 


30/00 


S6800 


B6900 


saeoo 


478 


4/S- 


97 3 


23/ 


44.3 


3B.O 


377 


■404 


40/ 


0020/ 


/300 


/230 


3/300 


59/00 


904 00 


e/ /oo 


300 


30 


73 8 


9/0 


62.1 


49 Z 


6/ 2. 


72 5 


6/3 


.00307 



CD Gage Lines 
Average T/7/c/<ne55> of To/ye .083 in Area of ^Gct/ar) /.A72sg.ir> 



Water /=^essure 
Ibperspn 


A/^/allood 

due to 

VPressure 


ifachine 
7-Oad 
pourjds 


Totai 
AtiolOxd 
paundi 


Ay. Axial 
VnitSnesi 
Ibpery^in. 


ffead/ng on Gage line 


Differences 


Ay. 
Diff. 


Ay 

Unit 

€k>r,get/on 


CD/ 


CD2 


COS 


CD4 


CD/ 


CD2 


CDS 


C04 


GogeR'd-g. 


Corrected 


/oo 


/OO 


2430 


s/oo 


7500 


5/00 


eaz 


44.9 


9S.0 


40.9 




















300 


300 


7290 


/'^/OO 


2/400 


i4600 


64/ 


39.5 


39 & 


34 9 


4.1 


54 


5.2 


6.0 


S.2 


.00026 


SOO 


500 


/2/50 


230S0 


35200 


23900 


S9.0 


347 


B'f.O 


30.9 


9 2 


/O.Z 


//o 


/OO 


/o. 1 


. ooos/ 


7 00 


7 00 


/7 000 


32/00 


■49iOO 


33300 


53.0 


290 


789 


2SO 


iS.Z 


15.9 


/6./ 


15.8 


758 


.0OO79 


SOO 


goo 


Z/900 


4/ /OO 


63O0O 


4-2900 


473 


23 Z 


74.0 


/9.4 


20.9 


2/7 


2/0 


2/5 


2/3 


.00/07 


9SO 


950 


23/00 


■4 3400 


6 6 500 


4- 5/ 00 


■46/ 


22.0 


73.0 


/G.3 


22.1 


229 


220 


22.6 


22.4 


.00 //Z 


/OOO 


/ 000 


24300 


45600 


69900 


47500 


■439 


20.0 


7/3 


/7.0 


24. Z 


?49 


23 7 


23 3 


Z4.Z 


. 00/21 


/ oso 


/OSO 


25500 


4-7900 


734-00 


49700 


410 


/7/ 


68 3 


/S.2 


272 


27B 


2S7 


251 


26.6 


.00/33 


//oo 


/oso 


2 6 500 


50/00 


76600 


52200 


388 


14/ 


67 


73 9 


e3* 


308 


260 


270 


23 


. 00/45 


//so 


//■40 


z 7700 


524O0 


eo/oo 


54400 


3S4 


/// 


es.o 


//.o 


3ZB 


33.7 


300 


29.1 


3/4 


00/57 


/ 200 


/J90 


29 ZOO 


S45O0 


B3500 


56800 


32 


74 


62.5 


9.0 


36Z 


37s 


32 5 


3/e 


34 5 


.00/73 


/2SO 


/240 


30/00 


56800 


86900 


59000 


26.9 


4.0 


SSO 


4 S 


4/3 


40 7 


40.0 


36.4 


39.6 


.00/98 


/3O0 


1290 


3/300 


59/00 


90400 


e/5oo 


8.0 


860 


47Z 


90 


60Z 


58.9 


47.8 


50 9 


54 S 


.00273 



BECKER STEEL UNDER BIAXIAL LOADING 33 

that the distinction between yield point and p-limit is very slight and 
that it really makes no material difference which is used. But a glance 
at the stress-strain diagrams in Fig. 13 to 16, will show that in some 
cases the modulus of elasticity changes and that the diagram consists 
of a broken line instead of a straight line nearly up to the yield point. 
This fact, due to the lack of isotropy in the material and to the mechan- 
ical work done upon it, makes it difficult to get consistent results by using 
the p-limit as a criterion. When the material has been cold worked, 
the stress-strain diagram often curves away from a straight line slowly 
and the exact point of departure is not easily located. Special treatment 
of the material usually affects the yield point in the same way in differ- 
ent specimens, but not the p-limit. 

The use of rupture or ultimate strength as a criterion of the strength 
of ductile materials still persists in the case of simple stresses, and 
specifications ordinarily require that the ultimate strength of the ma- 
terial shall have a certain value. But this is an indirect measure of 
the toughness rather than of the strength, and in the best specifications 
the yield point (or elastic limit as it is frequently but incorrectly called) 
is specified as well. Conditions at rupture give no indication of those 
existing at the yield point and whatever value a knowledge of the condi- 
tions attending rupture in a ductile material may have, no conclusions 
can be drawn from them which may safely be applied to the period pre- 
ceding the yield point. As engineering design deals principally with 
stresses within the yield-point stress, rupture cannot be considered as 
the criterion, even though Bridgman* in his tests on thick cylinders 
uses it and decries the use of the yield point. When the distribution 
of stress is unknown and no extensometers are used to measure strains, 
rupture is the only criterion available. 

For ductile material that has not been worked cold, the stress-strain 
diagram shows a very decided change in character when the material 
passes the yield point. When the material has been cold-rolled or cold- 
drawn, the yielding is more gradual and the curve, instead of breaking 
sharply, departs more gradually from a straight line. If the specimen 
of the cold-rolled or cold-drawn material is tested in simple tension with 
an extensometer, and the load is slowly but steadily applied, the roll of 
the curve is apparent a short time before the yield point is registered 
by the drop of the beam. 

As all the investigations hereinafter described were made with in- 
struments to measure the strains, some criterion must be adopted that 



•Phil. Mag., July, 1912. 



34 



ILLINOIS ENGINEERING EXPERIMENT STATION 



is applicable to a stress-strain diagram. The first deviation from a 
straight line (p-limit) is an indefinite point to locate, and, after con- 
sidering everything that has been noted above, the method proposed by 
the late J. B. Johnson was adopted. This is called by him the "apparent 
elastic limit," although it is here taken as the yield point. This method 
empirically locates a point at ^hich there is evidently some plastic 
action and furnishes a very convenient method for comparison of results. 
It is defined as the unit-stress at which **^the rate of deformation is 50 
per cent greater than it is at zero stress." Fig. 12 shows the application 
to a stress-strain diagram. Let B E be a stress-strain diagram drawn 
in the usual manner. Then A B is the angle determining the slope 




Unif5tro/n 
Fig. 12. Stress-Strain Diagram Showing Johnson's Apparent Elastic Limit. 



at zero stress. At any point K lay off horizontally a distance K F equal 
to 1.50 times K B. Then F is the slope 50 per cent greater than the 
slope at zero stress. A parallel to F drawn tangent to the curve B E, 
locates the point of tangency L and the corresponding stress is the yield- 
point stress. 

23. Strength. — In the tabulation of the results of the tests of tubes 
under biaxial stress, the average of the strains measured on the four 
gage lines intersected by any cross section was taken as the strain at 
that section of the tube. Thus, the strains for the axial gage lines, 
ABl, AB2, AB3, and A B4 of a tube (for notation see page 23 and 



BECKEK STEEL UNDER BIAXIAL LOADING 



35 



60000 



50000 



40000 



i 

«f5 



30000 



20000 




10000 



Unit 5 train - One Division =. 0002 — -OOOl- — 

L denotes yield point 

Fig. 13. Stress-Strain Diagrams for Tube No. 3, Series 1. Ratio of Circum- 
ferential TO Axial Tension, 0.94. 



70000 



60000 



50000 



40000 



<^ 



30000 



20000 



10000 




Unit dfroin - One Division -0002 



L denotes yield point 

Fig. 14. Stress-Strain Diagrams for Tube No. 4, Series 1. Ratio of Cir- 
cumferential TO Axial Tension, 0.69. 



36 



ILLINOIS ENGINEERING EXPERIMENT STATION 



Fig. 9) were averaged, and this average is taken as the strain of the A B 
gage lines of that tube. Likewise for the B C and C D gage lines. For 
the circumferential gage lines, 1-2-A, 2-3-A, 3-4-A, and 4-1-A were 
averaged; that is, the four gage lines made a complete traverse of the 
circumference. There are then three sets of average results for the 



JOOOO 
^20000 
^ 10000 



Unit dtrain-One Div/sidP'O.OOO^ -^.0002-' — 

L denotes yield point 

Fig. 15. Stress-Strain Diagrams for Tube No. 2, Series 3. Ratio of Circum- 
ferential Tension to Axial Compression, 0.20. 



o- 


— 




-^ 








^< 


Pv" 


>i 






































k, 


















i^ 


\ 






Ps 


k 


X 


,< 








CifTumferen/fo/ Gage Lines 

1 1 1 J II 1 










/ 


^/C/£^ 


>/G 


oge 


'Lin 


es' 


\ 


c^ 




^ 




^. 














































K^ 


% 




V 


^■^ 














































% 




\ 


S 


\ 


\ 




elif^^'^ 


\J-\ — 


le lines CGo 


7el 


ines 


DGage 




'J 




















\ 






\ 




s 


^ 


> 


r 


A 


^ 




Z 


r 



axial gage lines and four for the circumferential gage lines of each tube. 
The curves formed from these average results (see Fig. 13 to 16 for 
samples) were then used to obtain the general average results for each 



dOOOO 




Unit 5/ro/n- One Di/is/'on = aoO0^ 



L denotes yield point 

Fig. 16. Stress-Strain Diagrams for Tube No. 5, Series 3. 
ferential to Axial Tension, 0.30. 



Ratio of Circum- 



of the tubes. That is, the general average results for the circumferential 
strains represent the average obtained from all the circumferential gage 
lines in any one tube, and the general average results for the axial strains 
the average obtained from all the axial gage lines. The only exception is 
in the case of tube No. 1, Series 1, where the averages of the A B gage 
lines are omitted in the general average. Each general average curve 
represents the average results of twelve axial gage lines or sixteen circum- 



BECKER — STEEL UNDER BIAXIAL LOADING 



37 



ferential gage lines. These general average stress-strain diagrams are 
given in Fig. 17 to 19. 

The yield-point stresses are quite uniform for the different sets of 
gage lines and in close agreement with those of the general average 
curves. Because of this uniformity, the use of the general average 
curves as a basis of comparison seems justified. The circumferential 
strains are plotted with the apparent circumferential tensile stresses as 
ordinates, except in the case of the tubes where no internal pressure was 
applied. In these cases the ordinates are the axial stresses, so that it 



eoooo 




Unit strain -One Dmsion=0.000^ *^.000^ 

Note- Figures fol/om'ng fut?e numbers on curves Math Theory offJas 

inc/icate rath of circumferential to ami stress. fguii^.SimpJe Stress 

L denotes yield point 

Fig. 17. Stress-Strain Diagrams Showing General Averages for Series 1. 
Tension With Tension. 



is easy to determine Poisson's ratio, which is given in Fig. 19 by the 
ratio of abscissas, corresponding to the same stress, on the two curves 
of tube 4, such as r to r'. 

If diagrams are drawn ha^dng the yield-point unit-stresses as 
ordinates and the ratio of the circumferential tension to axial tension 
or axial compression as abscissas, a comparison can be made with the 
results reached by the different theories. For the combination of ten- 
sion wdth tension, the maximum stress theory and the maximum shear 
theory demand that the yield-point stress shall be constant for all ratios. 



38 



ILLINOIS ENGINEERING EXPERIISIENT STATION 



Mohr's theory and the internal friction theory have the same require- 
ments; Wehage's theory demands a reduction in the yield-point stress- 
and the maximum strain theory demands an increase in proportion to- 
the increase of stress ratio. 

For the combination of compression with tension the maximum 
stress theory demands that the yield-point stress shall be constant for all 




Unit Strain - One Division = 0. OOOB 

Nofe- figures follomng tube number:^ on curves 

indicate ratio ofcircumferenfiol to a/'ial stress. 



— ^0003* 

Iftiat/iTiieoryoff/os. 

£qui\r. Simple Stress. 



L denotes yield point 

Fig. 18. Stress-Strain Diagrams Showing General Averages for Series 2. 
Tension With Tension. 



zoooo 




Note,- f'igures following tube numbers on curves 
indicate ratio ofcircumrereniiol fo axial d tress. 



Unit 5tmin- One Division =0.000^ —*^og^t< 

Mali). Theory offlos. 

Equiv. Simple Stress. 

L denotes yield point 

Fig. 19. Stress-Strain Diagrams Showing General Averages for Series 3. 
Compression With Tension. 



stresses^ while the maximum strain theory, the internal friction theory,, 
the maximum shear theory, and Mohr's theory demand a decrease in 
the yield-point stress as the stress ratio increases. 

What is the law that governs ? Referring to the stress-strain curves 
of the general averages of the axial gage lines, Fig. 17, 18, and 19, it 
will be seen that for Series 1 and 2 as the stress ratio increases the yield- 
point stress rises unmistakably until the value of the stress ratio of 



BECKER — STEEL UNDER BIAXIAL LOADING 



39 



O.50 is reached. Beyond this the yield-point stress remains constant, 
no matter what the stress ratio. For Series 3 the yield-point stress 
steadily diminishes as the stress ratio increases. 

Since for the case of compression combined with tension all the 



50000 




.00/ 
UnifSrroin 

^ denotes yield point 

"Fig. 20. Tension Tests of Small Specimens From Tubes of Series 1, 2 and 3. 
Results of Ten Tests for Each Series. 



^0000 



i 



10000 



I 











I 

1 








































^ 


^ 


— © 


x>- 




— t: 


-^ 


o- -o- 












<-' 


^rr 


"^ 
























/ 


\ 




























' 


^ 






























/ 






























/ 




1 

1 


























j 


V 






























/ 






























1 


.. 































OOf 



.005 



.002 
Unif Detrusion 

Fig. 21. Torsion Test of Tube No. 6, Series 3. 



.00^ 



theories except one demand a decrease of the yield-point stress as the 
.stress ratio increases, while for tension combined with tension the 
maximum strain theory is the only one which calls for the increase that 



40 



ILLINOIS ENGIISrEERING EXPEEIIkCENT STATION 



has been observed. Series 1 and 2 will be discussed first and the 
results of Series 3 compared with the conclusions drawn from the 
results of Series 1 and 2. 

10000 



60000 



!;^ 50000 






.;§§ 40000 



30000 



20000 




'qr/ng5/re5^for ffaf/'o of 0.50^ 
1 - (Ouesrslai^A I 



ai C? 0.3 04 0.5 0.6 0.7 0.6 OS W 
Ratio of CIrcumferenfJa/ Tens/on /o/lx/a/ Tension 



Fig. 22. Diagram Giving Yield Point Stresses and Stress Ratios for Series 

1 AND 2. 



In Fig. 22 the yield-point stresses of Series 1 and 2 are plotted 
against the ratio of circumferential tension to axial tension. The line 
of the maximum strain theory is then drawn through the yield-point 
stress determined in simple tension (stress ratio zero), the inclination 
being determined by Poisson's ratio (0.334). For Series 1 the yield- 
point stress was taken from the test of tube No. 5 and for Series 2 the 



BECKER — STEEL UNDER BIAXIAL LOADING 41 

average of the tension tests of twenty small specimens cut from tubes of 
Series 2 was taken.* 

The determination of Poisson's ratio is discussed on page 46. A line 
of constant yield-point stress is drawn which best fits the experimental 
points for stress ratios of 0.50 or above. It is seen that the line of the 
maximum strain theory fits the experimental points up to its inter- 
section with the line of constant yield-point stress, and that thereafter 
the line of constant yield-point stress well fits the points. This line of 
constant yield-point stress may also be a line of constant shearing stress. 
If, as Fig. 22 seems to indicate, tension ceases to be a governing factor 
and the shearing stress becomes dominant, two things must be true for 
the line of constant yield-point stress : 

(a) The shearing unit-stress must actually reach the shearing 
yield-point stress as determined by tests in pure shear, and 

(b) Since the shear is one-half the maximum principal stress, this 
maximum principal stress must remain a constant. 

The first condition is important only in so far as showing that the 
shearing yield-point stress must be greater than one-half the tensile yield- 
point stress; otherwise the shear would be dominant at all times. The 
latter is the contention of the maximum shear theory. Counting com- 
pression a negative tension and with the principal unit-stresses num- 
bered in the order of their magnitude, p^, po, p^, the' criterion for shear- 
ing stress is: 

Shearing unit-stress = % (;?i — Ps), 
but as the third principal stress is zero, this reduces to % p^. The water 
pressure inside the tube does not constitute a third principal stress 
(compressive), for all the readings of the strains were taken on the out- 
side of the specimen where the third principal stress was undoubtedly 
zero, if the atmospheric pressure is neglected. 

The maximum shear theory carried to its logical conclusion re- 
quires that the yield-point stress of the material subjected to two 
stresses of like sign at right angles shall not vary from that reached in 
simple tension, for the shear is the determining factor at all times. 
If the theory holds in this form, a horizontal line drawn through the 

*The tension test of tube No. 9, Series 2, the first test made, did not furnish the 
necessary data on account of an unexpectedly low yield point. It is thought that the use 
of the yield-point stress obtained from the average curve for the specimens from the tubes 
of Series 2 (see Fig. 20) is justified because the break of the curve of the small specimens 
from the tubes of Series 1 agrees closely with the break in the curve obtained from tube 
No. 5 of that series (axial load only), 42,500 lb. per ?q. in. and 43,000 lb. per sq. in. 
respectively. The yield-point stress obtained from the average curve of the specimens from 
the tubes of Series 2 (21,500 lb. per sq. in.) has been taken as the yield-point stress in 
simple tension of Series 2 and the value of Poisson's ratio obtained from Series 1 has been 
used for Series 2. 



42 



ILLINOIS ENGINEERING "EXPERIMENT STATION 



yield-point stress in simple tension should pass through all the points. 
Instead, it touches only the initial point. Experiments* have shown 
that for ductile material the ratio of the shearing yield-point stress, ob- 
tained by torsion tests, to the tensile yield-point stress varies with the 
material, but usually lies between 0.55 and 0.65, in the majority of 



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0.1 0.2 0.3 04 0.5 06 07 0.8 0.9 /O 

Ratio Of Cinoumferenf/o/ Tension fo/J/(/a/ Compress/on 
Fig. 23. Diagram Giving Yield Point Stresses and Stress Ratios for Series 3. 



tests ranging near 0.60, which is the commonly accepted value. A few 
tests show a ratio less than 0.50, but they are relatively small in number. 
With a ratio of 0.60, the shearing yield-point stress line would lie above 
the line through the yield-point stress in simple tension by an amount 
equal to 0.20 of the latter stress. The exact location of the line will 

*L. B. Turner, Engineering, London, February 6, 1900. 



BECKER STEEL UNDER BIAXIAL LOADING 43 

vary with the material, but as long as the ratio of the yield-point stresses 
is above 0,50, there is the hiatus between this condition and that de- 
nianded by the above form of the maximum shear theory. 

The horizontal line through the experimental points in Fig. 22 is 
evidently the limit of the shearing strength. It corresponds to a ratio 
of shearing yield-point stress to tensile yield-point stress of 0.59 for 
Series 1 and 0.62 for Series 2, which values agree well with the majority 
of experiments. 

These tests indicate that there are two laws covering the case of 
combined stress when the stresses are both tension and act in two direc- 
tions at right angles. Apparently the point at which the change in law 
occurs depends upon the ratio of the yield-point stress in shear to that 
in tension and the change from one law to the other may occur at different 
ratios of the principal stresses for different materials. It is important to 
establish this ratio of yield-point stresses, for if it is not approximately 
constant the use of combined stress formulas will require a knowledge 
of such a ratio for all materials. 

Before discussing Series 3, the two laws just referred to will be 
applied to the other combinations of stress and a comparison made with 
the maximum stress theory, the maximum strain theory, and the maxi- 
mum shear theory. Assuming the ratio of the shearing and tensile 
yield-point stresses to be 0.60 and the tensile and compressive yield-point 
stresses equal, the co-ordinates of the rectangle A B C D (Fig. 24) repre- 
sent the maximum stress theory, the rhombus Q K J L the maximum 
strain theory, and the figure A K^, B C L^, D A the maximum shear 
theory. The line A M K,, N B represents the two laws in the tension- 
tension quadrant, while B R S C represents them in the tension-compres- 
sion quadrant. The lines M Kg and KoN are parallel to the axes and at 
such a distance from them that the ordinate of M K2 and the abscissa 
of K2N are each 1.20 times A or OB, the tensile yield-point stress. 
R S is parallel to B C and at such a distance from it that one-half the 
sum of the ordinate and abscissa of any point between R and S is equal 
to 0.60 of B or C. The construction of the other two quadrants is 
such that the figure is sjTnmetrical about the bisectors of the quadrants. 
The diagrams, Fig. 22, showing the comparison of theory and experi- 
ment for Series 1 and 2 correspond to A M Kg ^^ "the tension-tension 
quadrant. 

In Fig. 23 the yield-point stresses of Series 3 are plotted as ordinates 
and the stress ratios of circumferential tensile stress to axial compressive 



44 



ILLINOIS ENGINEERING EXPERIMENT STATION 



stress as abscissas. Before discussing the various theories in connection 
with the experimental results, the starting points of the theoretical lines 
must be fixed. The maximum shear theory demands the same yield- 
point stress in tension and in compression; the maximum strain theory 
and the maximum stress theory do not. From the average curve of ten 
specimens cut from a ten-inch remnant of the original tubing from 



Compression 



^^ 




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A 

:7 



/Abscissa- 




"i 
I 

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I 



Fig. 24. Representation of Yield Point Strengths for Combined Stresses 

According to the Maximum Stress Theory, the Maximum Strain 

Theory, and the Maximum Shear Theory. 



which the tube specimens of Series 3 were cut, (Fig. 20) the tensile yield- 
point stress was found to be 24,000 lb. per sq. in. The compressive 
yield-point stress obtained from tube Ko. 4 (no internal water pressure) 
was 26.250 lb. per sq. in. With the demand of the maximum shear 
theory for equal yield-point stress in tension and in compression it seems 
correct to take as the initial point of the line of that theory the average 
of these values, or 25,100 lb. per sq. in. The lines of the maximum 
strain theory and of the maximum stress theory were also drawn through 



BECKER — STEEL UNDER BIAXIAL LOADING 45 

this average value of the yield-point stresses and compressive yield-point 
stresses The use of this average value is believed to be justified by 
the observed fact that the yield-point stress of low-carbon steel in tension 
is found in nearly all cases to have the same numerical value as the 
yield-point stress in compression. In determining the line for the 
maximum strain theory a value of Poisson's ratio of 0.395 was used. 
This value was obtained from the test of tube No. 4, Series 3 (Fig. 19). 
A line of constant shearing stress has been drawn through the yield- 
point stress (14,000 lb. per sq. in.) obtained from the torsion test, 
since simple torsion produces tensile and compressive stresses of equal 
intensities and hence corresponds to a stress ratio of unity (see Fig. 21). 
The lines of internal friction theory and of Mohr's theory practically coin- 
cide with the maximum shear theory. 

An inspection of Fig. 23 shows that for Series 3 as well as for 
Series 1 and 2, the experimental results follow the maximum strain 
theory up to a certain stress ratio and then follow a line of constant 
shear which is the maximum shear developed. The ratio of the shear- 
ing yield-point stress from the torsion test to the average of the tensile 
and compressive yield-point stresses is 0.56. The question of the 
neglect of water pressure as a third stress does not enter in this series, 
for taking the stresses in the order of their magnitude the compression 
due to the water pressure becomes intermediate between the circum- 
ferential tension and the axial compression, so that the maximum shear- 
ing stress is equal to one-half the sum of the axial compressive stress 
and the circumferential stress. This series leads to the same conclusions 
as the other two, although the ratio of the shearing and tensile yield-point 
stresses is somewhat lower. 

The net result of this investigation as it affects the strength of steel 
under combined stress in two directions at right angles to each other — 
biaxial loading — is that instead of a single law, whatever its nature, as 
has heretofore been assumed, there are two distinct laws governing the 
strength of the material, each law dominant within its limits. These 
two laws are the maximum strain theory and the maximum shear theory ; 
the first governs until the shearing yield-point stress of the material is 
reached, after which the shear theory holds. The exact point of the 
change from one law to the other depends upon the ratio of the shearing 
yield-point stress to the yield-point stress in simple tension and com- 
pression. 

24. Stiffness. — Although strains have been measured in many tests 



46 ILLINOIS ENGINEERING EXPERIMENT STATION 

heretofore made, no attempt seems to have been made to determine the 
law of stiffness. It has been taken for granted that the deductions of 
the mathematical theory of elasticity, as embodied in St. Tenant's theory, 
hold, or else no attention has been paid to strains except as related to 
the strength of the material in the determination of the yield point or 
so-called elastic limit. The weakness of the mathematical theory of 
elasticity lies in its generalization of Hooke's law and the neglect of the 
temperature changes, so that the strains obtained in tests of isotropic 
materials will only closely approximate the computed values. The effect 
of shear in producing strain has been neglected and is small before the 
yield-point stress is reached, but the variation of shearing strength in 
different directions throughout the specimen, the possibility of a change 
in Poisson's ratio with increasing stress, the possibility of a different 
Poisson's ratio and modulus of elasticity with and across the direction of 
rolling or drawing, enter to complicate the problem. The material ex- 
perimented upon is not the isotropic substance assumed in the theory. 
Lines have been drawn on the stress-strain curves of the general aver- 
ages of the axial gage lines, Fig. 17, 18, and 19, giving the strains as 
computed by the mathematical theory of elasticity using the values of 
Poisson's ratio* and modulus of elasticity obtained from tests in simple 
tension and in compression. These lines agree quite closely with the 
observed values except in the case of tube No. 1 of Series 1, and tube 
No. 7 of Series 2, the former showing lower strains and the latter 
greater strains than the computed values. Apparently within the range 
of application of the mathematical theory of elasticity, where E is con- 
stant, the strains follow the theory with sufficient exactness to say that 
the theory holds. Lines have also been drawn to represent the strains 
corresponding to a simple tensile or compressive stress equal to the 
greater principal stress. 

For the circumferential lines there have been drawn on the stress- 



*The values of Poisson's ratio for the tubes tested in simple compression and in simple 
tension were obtained by dividing the circumferential unit-strain taken from the general 
average curves of these tubes (which is the same as the diametral unit-strain) by the 
corresponding axial unit-strain. For Series 1 this ratio for tube No. 5 is 0.334; for 
Series 3, obtained from tube No. 4, it is 0.395. The modulus of elasticity of Series 1 is 
27,200,000 lb. per sq. in., and for Series 3 it is 29,500,000 lb. per sq. in. An examination 
of the axial and circumferential stress-strain diagrams of tube No. 5, Series 1, Fig. 17, and 
of tube No. 4, Series 3, Fig. 19, shows that in the first case (tension) Poisson's ratio 
remains practically constant, diminishing about 6 per cent after the yield-point stress has 
been passed, but that in the second case (compression) this latio increases to almost 0.50 
after the yield-point stress has been passed. There is no reason why Poisson's ratio should 
be constant for all kinds of steel, and it may well be that tension and compression tests 
on the same material will show different results. It is not known what the effect of the 
hollow specimen is in changing this ratio for tension or compression tests, but it is 
thought that the method used to obtain Poisson's ratio is accurate and reliable. It is to be 
noted that for the compression tests the value of both Poisson's ratio and the modulus of 
elasticity are higher than for the tension tests. 



BECKER — STEEL UNDER BIAXIAL LOADING 47 

strain curves of the general averages, Fig. 17, 18, and 19, lines giving 
the strains computed by the mathematical theory of elasticity and also 
the strains accompanying simple tensile stresses equal to the circum- 
ferential stresses. The values of Poisson^s ratio and the modulus of 
elasticity are taken the same as for the axial lines. The lines of the 
mathematical theory of elasticity do not fit as well as in the case of the 
axial strains. It can be seen that to fit the experimental points of 
Series 1 and 2, it is necessary to use a higher value for both the modulus 
of elasticity and Poisson's ratio, the latter requiring the greater change. 
An increase in Poisson's ratio will increase the strains of tube No. 1 
and lower those of the other tubes of these two series. An increase of 
the modulus of elasticity will diminish all the strains proportionally. 
It will be recalled that the value of Poisson's ratio obtained in compres- 
sion tests was high, 0.395. For Series 3, where Poisson's ratio is higher 
than for Series 1 and 2, the modulus of elasticity alone need be increased. 
With a higher modulus the computed circumferential curves fit the ex- 
perimental curves quite closely except for tube No. 5. There is a strong 
probability that both Poisson's ratio and the modulus of elasticity vary 
in the two directions with and across the direction of drawing. It is 
scarcely probable that the law changes, and the close agreement between 
the computed and observed values for the axial strains gives strong sup- 
port to the belief that all the strains follow the requirements of the 
mathematical theory of elasticity. The indications are that the modulus 
of elasticity and Poisson's ratio may be different in different directions 
throughout the steel, in much the same way that Bauschinger has shown 
that the shearing strength of rolled steel varies in different directions. 
In Series 1 and 2, Fig. 17 and 18, for the tubes tested with a stress 
ratio of 0.92, the yield-point stress in the circumferential direction was 
practically the same as the yield-point stress in the axial direction, but 
the circumferential curves show a more sudden yielding of the material. 
In Series 3, Fig. 19, for the tube tested with a stress ratio of 0.90, the 
circumferential yield-point stress was lower than that in an axial direc- 
tion. All the circumferential stress-strain curves of Series 3 show a 
sharp, sudden break when the yield-point stress in the axial direction 
is reached, no matter what the circumferential stress was. Granting 
that for Series 3 the value of Poisson's ratio increases to 0.50 above the 
yield-point stress, this is not sufficient to account for the great increase 
in the strains. It must be that the shearing stresses, which have passed 
the shearing yield-point stress, produce shearing strains of sufficient 



48 ILLINOIS ENGINEERING EXPERIMENT STATION 

magnitude to account for this increase in the circumferential strains. 
This explanation is more strongly suggested by the stress-strain curves 
of Series 1 and 2, where Poisson's ratio remains nearly constant. The 
circumferential stress-strain diagrams that continue to show an increase 
in strain after the axial yield-point stress has been passed are those 
from the tubes whose axial yield-point stresses lie on the line of constant 
shear of Fig. 22. Those that do not show an increase at this time are 
from the tubes whose axial yield-point stresses lie on the line of the 
maximum strain theory. 

That the shearing strains accompanying the axial stress can affect 
the circumferential strains is shown by the stress-strain diagrams for the 
circumferential lines of tube No. 4, Series 1, Fig. 14. The circum- 
ferential stress-strain diagram continues straight for a short distance 
after the yield-point stress has been passed in the axial direction, the 
circumferential stress corresponding to the axial yield-point stress being 
34,500 lb. per sq. in.^ approximately. This is during the stage inter- 
mediate between the elastic and plastic conditions. When the axial curve 
breaks sharply, the circumferential curve changes direction also. If 
Poisson's ratio were the only factor, all the diagrams, with the possible 
exception of those of tubes N'o. 3, 6, and 8, where a high stress ratio was 
used, would show diminishing strains with increasing stress after the 
yield-point stress in the axial direction had been passed. This means 
that the tendency to reduce the diameter of the tube, due to the rapidly 
increasing axial strains, would be greater than the tendency to increase 
the diameter produced by the increase of the water pressure. But the 
curves of tubes 'No. 2 and 4 show an increasing strain (increasing tube 
diameter) even though the circumferential stresses were well below the 
yield-point stress of the material. These two tubes are the ones whose 
yield-point stresses lie on the line of constant shear. Fig. 22, and with- 
out the assistance of the shearing strains in producing circumferential 
strains, the curves of these two tubes would show a diminishing circum- 
ferential strain as the circumferential stress increased after the axial 
yield-point stress had been passed. The shear which causes yielding in 
an axial direction is on a different plane from that causing jdelding in 
a circumferential direction. The former shear acts along a plane which 
passes through the direction line of the circumferential tension and cuts 
the axis of the tube at an angle of 45°. The latter shear acts on a plane 
which passes through the direction line of the axial tension and is 
parallel to the axis of the tube making an angle of 45° mth the direc- 



BECKER — STEEL UNDER BIAXIAL LOADING 49 

tion line of the circumferential stress. These shearing stresses are of 
different magnitudes, according to the ratio of the stresses, and each is 
equal to one-half the principal stress cut by its plane at an angle of 45°. 

Apparently the strains follow the requirements of the mathematical 
theory of elasticity for all stress ratios, but there may be different values 
of Poisson's ratio and the modulus of elasticity for the axial and circum- 
ferential directions. After the yield-point stress in one direction has 
been passed the shearing strains have a considerable influence upon the 
deformations in the second direction. 

25. Comparison With the Methods and Results of Other Investi- 
gations. — Attention is called to several points of difference between the 
method of investigation here recorded and the methods used by others. 
The greatest difference lies in the use of a portable extensometer to meas- 
ure strains, the strain gage, whereby a large number of measurements 
were taken, both along the specimen and around it. Previous investiga- 
tions used a fixed extensometer which measured strains along one or 
two gage lines, or, in some cases, used no strain measurements. No 
assumptions of uniform stress distribution were made, in the present 
series, for the strain gage records the variations and the gage length can 
be varied to suit the needs. With readings taken on a large number of 
gage lines for every load increment, a certain positiveness of result is 
attained which is impossible with attached instruments and few gage lines. 
Local effects are thus minimized. Another difference lies in the larger 
size of the specimens tested and in the smaller ratio of thiclmess of tube 
wall to diameter. Because of the form of specimen and the method of 
applying load, the stress was nearly uniform throughout the specimen; 
there was no "helping" effect by understressed material, no point of 
maximum stress to be located. The use of Johnson's apparent elastic 
limit method for determining yield-point stress gives a definite point for 
comparison. 

An attempt was made to keep a definite ratio between circum- 
ferential and axial stresses throughout the test of each tube, so that com- 
parison might be made later for these ratios. As far as possible, it Avas 
intended with a set of specimens cut from a given length of tubing to 
cover the entire range of stress ratio within tension-tension or compres- 
sion-tension quadrants. The* experiments reported by others and re- 
ferred to in this section show generally a haphazard ratio of stresses, 
and the loads used were such that a definite stress was produced in one 
direction and then the other stress was increased until yielding took place. 



50 



ILLINOIS ENGINEERING EXPEEIIIENT STATION 



The earliest important investigation of this subject was that reported 
by J. J. Guest* in 1900. The tests were made upon small steel, copper, 
and brass tubes about 114, in- outside diameter and varj'ing in thickness 
from 0.025 in. to 0.034 in. Tests were made in combined torsion and 



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0^ 04 0.6 0.6 W 

lim)//ng lenaile orCompressive 3fre55 (/Jny Unifs) 

Fig. 25. Relation Between Shearing Stresses Due to Torsion and the Ten- 
sile OR Compressive Stresses Due to Axial Load or Bending. 

axial tension, in torsion and circumferential tension, and in axial and 
circumferential tension. The strains were measured by a two-point 
extensometer, and although it was attached to the outside of the tube, 
the full hydrostatic pressure was counted as a third principal stress 
(compressive). Other than the tests on tube No. 1 of Guest's investi- 



♦Phil. Mag., 1900. 



BECKER — STEEL UNDER BIAXIAL LOADING 51 

gation, there are but two tests where the stress ratio of circumferential 
tensile stress to axial tensile stress is 0.50 or less, and test No. 1 and one 
of the others follow the maximum strain theory closely. The yield- 
point stress was used as the basis of comparison, and each test was 
carried just beyond the yield point. Criticism may be made of the 
repeated use of the same specimen, since the yield-point stress is raised 
by repeated loading beyond the yield-point stress of the first test. It is 
not stated whether the tubes were annealed between tests. The results 
are taken to justify the maximum shear theory, and in the main they 
do within the field investigated, since the majority of the tests had a 
stress ratio between 0.50 and 1.00 within which limits the shear theory 
undoubtedly holds. The tests also show that the maximum shear de- 
veloped is greater than one-half the yield-point stress in simple tension. 

Following Guest comes the work of C. A. M. Smith,* W. A. Scoble,t 
E. L. Hancock,^ and Wm. Mason* on bars and tubes in torsion and 
tension or compression and on small tubes in compression and internal 
pressure. All these results are used to justify the maximum shear theory 
which demands that the shearing yield-point stress is equal to one-half the 
yield-point stress in simple tension. With one exception, however, that 
of Scoble's tests reported in 1906, the maximum shear developed is 
greater than one-half the yield-point strength in tension, which, as noted 
above, was also found in Guest's tests. The majority of these tests — like 
Guesf s — are in the region where the stress ratio is greater than 0.50. 
These tests cover the entire four quadrants of combined stress. 

The tests of Professors Smith and Hancock will be shown on dia- 
grams similar to Fig. 25 (Fig. 26 and 27), in which the ordinates 
represent the shearing stress due to torque and the abscissas represent 
the tensile or compressive stress due to axial load or bending. The 
diagram of Fig. 25 will be discussed before the tests are taken up. The 
shearing stress is plotted to twice the scale of the tensile or compressive 
stress. If a circle with a radius equal to the tensile yield-point stress 
is drawn with as a center, it will represent the relation between the 
shearing and the direct stress which produces a combined stress causing 
yielding required by the maximum shear theory. It will be observed that 
the shearing yield-point stress must therefore equal exactly one-half the 
tensile yield-point stress. A circle with radius equal to the shearing yield- 
point stress obtained from tests in simple torsion (0.6 the tensile yield- 



*Inst. Mech. Engrs., 1909. 

tPhil. Mag., 1906 

JAm. Soc. for Testing Materials, 1905, 6, 7, 8. 



52 



ILLINOIS ENGINEERING EXPERIMENT STATION 



point stress) is shown; also, the ellipse representing the St. Yenant or 
maximnm strain theory, beginning at the tensile yield-point stress. The 
two laws as advanced in this bulletin require that the maximum strain 
theory hold to the intersection of the St. Venant ellipse and the circle 
for limiting shearing stress for ratio of 0.6, and that then the shear 



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Fig. 26. Results of Tests, by C. A. M. Smith. 



shall govern. Hancock's ellipse has been added to show how closely he 
came to the results here advanced. 

The results of Smith's and Hancock's tests have been plotted in 
Fig. 26 and 27. Both compression and tension have been plotted on 
the same side of the diagrams (symmetry permitting this), and the 
results of the different tests have been changed proportionally in order 
to compare them with a single set of theoretical curves. A comparison 
of the two laws herein proposed with the experimental results of these 
investigations show that the experimental results fit these laws better 
than the maximum shear theory which the tests were taken to prove. 

Fiff. 26 shows the results of C. A. M. Smith's tests on S. S. and 



BECKER — STEEL UNDER BIAXIAL LOADING Do 

A. D. steel. Professor Smith maintains that the shearing yield-point 
stress of steel is one-half the tensile yield-point stress within small limits 
and quotes Turner's tests* to prove his point. His own tests do not bear 
out his contention and Turner's tests show considerable variation, averag- 
ing about 0.54 for this ratio. Professor Smith's tests are examples of 
careful work, but the interpretation of the tests as an unqualified endorse- 
ment of the maximum shear law cannot be accepted. 

Mr. Scoble's tests seem to indicate that the shearing yield-point 
stress is lower than half the tensile yield-point stress. This result may 
possibly be accounted for by the way the shearing yield-point stress was 
located. This stress was taken at the intersection of the straight line 
of the elastic portion of the stress-strain diagram with a line drawn 
through the diagram beyond the yield point. Since a stress-strain curve 
for torsion breaks more quickly than a tension curve, it may be that 
the determination of the shearing yield-point stresses are affected by 
this. Scoble's method of measuring the bending moment by means of 
the deflection of the beam may be in error, for the law of deflection 
under the combined stress would be influenced by the very law he was 
seeking to determine. 

The results of Professor Hancock's testsf are shown in Fig. 27. The 
curves of the maximum shear theory and the maximum strain theory 
have been drawn as well as his ellipse. Hancock used the p-limit as his 
criterion. He alone of these investigators realized the shortcomings of 
the maximum shear theory and endeavored to remedy them by fitting 
an ellipse to the experimental results. The ellipse fits quite closely, but 
while it is a close approximation, it does not fit the results as closely 
as do the curves representing the two laws herein proposed. His ellipse 
is empirical, while the combination of the maximum strain theory with 
the maximum shear theory has a foundation in the theory of the strength 
of materials. 

Since torsion combined with compression or tension can be resolved 
into a case of tension combined with compression. Smith's and Hancock's 
tests fall in the fourth quadrant and show the applicability of the two 
laws there. 

Mason's tests on tubes in compression and internal pressure show 
that the maximum shearing stress developed is greater than the shear- 
ing stress developed in simple compression. The average of all his 

'Engineering, London, February 5, 1909. 

tProceedings of the American Society for Testing Materials, 1908. 



54 



ILLINOIS ENGINEERING EXPERIMENT STATION 



tests in which a constant stress ratio of one to one was used, gives this 
maximum shearing stress as 0.60 of the compressive yield-point stress. 
As all the tests had the one stress ratio, it is not possible to make a 
comparison with theories, But the point thus located falls on the line of 
the two laws. 

Minor inconsistencies are to be expected in experimental work of 
this nature, both on account of the variation in the material tested and 



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Fig. 27. Results of Tests, by E. L. Hancock. 



on account of the apparatus used. The number of tests made by these 
investigators is insufficient to establish completely any theory, but a care- 
ful study of the published data will lead to the conclusion that the two 
laws, the theory here advanced, conform more closely to the experimental 
results than any single law. 

26. Summary and Conclusions. — The following summary deals with 
the method of investigation and with the deductions which have been 
made from the data. As this is the first investigation of combined 
stress wherein a portable strain measuring instrument — such as the strain 



BECKER STEEL UNDER BIAXIAL LOADING 55 

gage — has been used, it is felt that considerable emphasis may be laid 
upon this fact. The size of the specimen is much larger than any here- 
tofore used. These conditions tend to give more trustworthy results. 
The experimental conclusions are : 

1. The use of a portable strain measuring instrument is a de- 

cided advantage since it makes it possible to take measure- 
ments on a large number of gage lines for each increment 
of load, obviating to a large extent the effect of local varia- 
tions in the test specimen. 

2. The use of large tubes with thin walls gives quite uniform 

stress distribution, the yield-point stress is more positively 
determined, and the effect of eccentricity of loading is less 
than with solid bars on account of the larger diameter of 
the tube. 

3. With large tubes the thickness of the tube walls can be accurately 

determined. 

4. Flat plates in cross bending give uneven distribution of stress 

and are not satisfactory for biaxial loading tests. 
The deductions which have been made from the experimental data 
are: 

5. With increasing values of the ratio of the biaxial stresses the 

yield-point strength follows the maximum strain theory until 
the value of the shearing stress reaches the shearing yield 
point, then the shearing stress controls according to a maxi- 
mum shear theory. There are thus two independent laws each 
dominant within proper limits instead of some single law as 
has heretofore been assumed. 
G. Because these two laws govern the strength of ductile materials 
under biaxial loading, the ratio for simple stresses of the 
shearing yield-point stress to the tensile yield-point stress is 
important. 

7. The stiffness follows the requirements of the mathematical 

theor}' of elasticity for all stress ratios, but the values of 
Poisson's ratio and the modulus of elasticity may be different 
in the two directions, with and across the rolling and drawing 
of the steel. 

8. The results of the tests reported by previous investigators con- 

form better to the two laws of strength than to any single 
law. 



56 ILLINOIS ENGINEERING EXPERIMENT STATION 

APPENDIX I. 

Bibliography. 

Basquin, O. H., The Circular Diagram of Stress and Its Application to the 

Theory of Internal Friction. Proc. Western Society of Engineers, Novem- 
ber, 1912, page 815. 
Bridgman, p. W., Breaking Tests under Hydrostatic Pressure and Conditions 

of Rupture. Philosophical Magazine, July, 1912, page 63. 
Crawford, W. J., The Elastic Strength of Flat Plates, an Experimental Research. 

Proc. Royal Society of Edinburgh, 1911-12, page 348. 
Cook and Robertson, The Strength of Thick Hollow Cylinders under Internal 

Pressure. Engineering, London, December 15, 1911, page 787. 
Editorial in Engineering, London, November 29, 1912, page 745. 
Guest, J. J., On the Strength of Ductile Materials under Combined Stress. 

Phil. Mag., July, 1900, page 69. 
Gulliver, G. H., Deformations in Metals. Proc. Inst. Mech. Engrs., February, 

1905, page 141. 

On the Cohesion of Steel and on the Relation Between the Yield Points 

in Tension and Compression. Proc. Royal Society of Edinburgh, 1907-8, 

page 374. 

On the Effect of Internal Friction in Cases of Compound Stress. Proc. 

Royal Society of Edinburgh, 1908-9. 

Internal Friction in Loaded Materials. International Association for 

Testing Materials, 1909, VIII. 

A New Experimental Method of Investigating Certain Systems of 

Stress. Proc. Royal Society of Edinburgh, 1909-10, page 38. 
Hancock, E. L., Results of Tests of Materials Subjected to Combined Stress. 

American Society for Testing Materials, 1905-6-7-8. Philosophical Magazine, 

October, 1906; February, 1908. 
Hatt, W. K., Moduli of Elasticity for Compound Stress. Proc. Purdue Society 

of C. E., 1897. 
Lanza, G., Applied Mechanics, 1910. 
Lawson and Capp, Thermo-Electric Indication of Strain as a Testing Method. 

International Association for Testing Materials, 1912, IXs. 
Love, A. E, H., The Mathematical Theory of Elasticity, 1904. 
Malaval, M., Notes on the Strength of Cylindrical Tubes Stressed to Rupture. 

International Association for Testing Materials, 1912, Xs. 
Mallock, a., The Extension of Cracks in an Isotropic Material. Proc. Royal 

Society of London, 1909, page 26. 

Some Unclassified Mechanical Properties of Solids and Liquids. Proc. 

Royal Society of London, 1912, page 466. 
Mason, Wm., Mild Steel Tubes in Compression and under Combined Stress. 

Inst. Mech. Engrs., 1909, page 1205. 
Matsumura, T., and Hamabe, G., Tests on Combined Bending and Torsional 

Strength of Cast Iron. Memoirs of the College of Engineering, Kyoto 

Imperial University, February, 1915. 
Mesnager, M., Deformations des Metaux. Comptes Rendues, Vol. 126, 1898, 

page 515. 



BECKER — STEEL UNDER BIAXIAL LOADING 57 

MoHR, O., Welche Umstande bedingen die Elastizitats grenze und den Bruch 
eines Materials? Zeitschrift des Vereins Deutcher Ingenieure, 1900, page 
1530. 

Rasch, Ew.^ Method for Determining Elastic Strength by Means of Thermo- 
Electric Measurements. International Association for Testing Materials, 
1909, VII3. 

ScoBLE, W. A., The Strength and Behavior of Ductile Materials under Combined 
Stress. Philosophical Magazine, December, 1906, page 533. 

Ductile Materials under Combined Stress. Philosophical Magazine, 1910, 
page 116. 

Tests of Brittle Materials under Combined Stress. Philosophical Maga- 
zine, 1910, page 908. 

Report on Combined Stress. British Assn. for the Adv. of Science, 
1913, page 168. 

Slater and Moore, Use of the Strain Gage in the Testing of Materials. Ameri- 
can Society for Testing Materials, Vol. 13, 1913, page 1019. 

Smith, C. A. M., Compound Stress Experiments. Inst. Mech. Engrs., 1909, 
page 1237. 

Some Experiments on Solid Steel Bars under Combined Stress. En- 
gineering, London, August 20, 1909. 

The Strength of Pipes and Cylinders. Engineering, London, March 5, 
1909, page 327. 

Guest's Law of Combined Stress. Engineering, London, April 23, 1909, 
page 545. 

Talbot and Slater, Tests of Reinforced Concrete Buildings. Bulletin 64, En- 
gineering Experiment Station, University of Illinois. 

Turner, C. A. P., Thermo-Electric Measurement of Stress. Trans. Am. Soc. 
C. E., 1902, page 26. 

Wehage, H., Die Zulassige Anstrengung eines Materials bei Belastung nach 
Mehreren.Richtungen. Zeitschrift des Vereins Deutcher Ingenieure, July, 
1905, page 1077. 

Williams, L, A Determination of Poisson's Ratio. Philosophical Magazine, 
1912, page 886. 



58 ILLINOIS ENGINEERING EXPERII^IENT STATION 

APPENDIX II. MATHEMATICAL TREATMENT. 

1. Stresses and Strains, — The analysis of stress and strain in elastic 
materials known as the mathematical theory of elasticity embodies the 
most complete and elaborate theory of the action of elastic bodies under 
stress. The following brief presentation of the mathematical theory of 
elasticity as it applies to the problem of the investigation follows largely 
the treatment of Love.* It will be desirable to outline briefly the work 
leading up to the derivation of the general equations of the mathematical 
theory of elasticity connecting stress and strain before taking up the 
derivation of the equations of stress and strain in a cylinder under 
internal pressure and an axial load. 

In the theory of elasticity the relations between three sets of 
magnitudes must be considered. 

1. The displacements of the points of the strained body. If the 
ordinary rectangular system of coordinates is used for reference, the 
displacement s of a point due to the strain is resolved into components 
u, V, w parallel respectively to the X, Y and Z axes. 

2. The strain components. Let c^, Cg, £3 denote the strains in the 
directions of the X, Y, Z axes, respectively; then 

6u dv dw 

ei= —,^2= — >^3=-T- (1) 

dx dy dz 

The components of shearing strain are defined as follows : 

dw dv du dw dv du 

^23= —-\- J- ^ ^31 = — + — J «12 = — + — (2) 

dy dz dz dx dx dy 

Here C23 denotes the shearing strain in the plane YZ, etc. Along 
with the strain components may be included the components of the 
rotation of an element of the body. If the displacement involves a rota- 
tion 0) of the element as a whole and this rotation be resolved into 
X, Y and Z components, then these components are given by the rela- 
tions 

dw dv du dw dv du 

^1= : — , «2 = T — , <^3 = : ~ (3) 

dy dz dz dx dx dy 

3. The stress components. The six stress components may be 
denoted by o-j, 0-2, 0-3 ; o-oa, o-guo-ij- o-^ is the stress in the direction of the 
Z-axis on a plane perpendicular to the X-axis; similarly for o-o and ag. 
0-23 is the stress in the direction of the F-axis over a plane perpendicular 
to the Z-axis; therefore it is a shearing stress. 

•The Mathematical Theory of Elasticity, A. E. H. Love, 1904. 



BECKER — STEEL UXDKR BIAXIAL LOADING 59 

The stress components must satisfy certain conditions of equi- 
librium, which are expressed by three equations of the following type 
(assuming that the body forces, such as gravity, may be neglected, and 
that the body is at rest). 

'-■ + '-^■-' + ^' = (4) 

OX oy 02 

The six strain components Cj, €2, etc., and the six stress components 

are connected by certain relations. Hooke's law, the linear relation 

between stress and strain, is the basis of these relations. Each stress 

component is taken as a linear function of the six strain components; 

thus 

a, = h,e, + h,e, + h,e, + h,e,, + h,e,, + I,.,, f ^^ ^ 

etc. 

A consideration of the work done in deforming a body leads to the 
conclusion tliat there must exist a so-called strain-energy function Y, 
such that 

6V dV 

o-i = — - , 0-2 = -— , etc. 

It follows that the function V must be a homogeneous quadratic 
function of the six strain components and must have therefore 21 terms. 
The number of coefficients apparently 36 in eq. (5) is thereby reduced 
to 21 by relations of the form a^, — h^, a^ = c^, a^ — d^, etc. ; that is, 

6 

V is the symmetric determinant of the quadric 2. ^\] ^ij. 

1 
If the body is isotropic, these 21 coefficients " can be reduced 
to two. Denoting by A one of these remaining coefficients and by fi one- 
half the diiference between the two coefficients, the following relations 
between stresses and strains are established: 

o-j = AA + 2/xci^ 

(To = AA -f 2/X€2 > (6) 

0-3 = AA -f 2ixeJ 

where a^, o-o and 0-3 are the stresses along the X, Y and Z axes respectively 
and Cj, €0 and €3 are the corresponding strains. A is the dilatation and 
is equal to the sum of c^, Cj and €3. 

Let — = Poisson's ratio 
m 

E = the modulus of elasticity. 



60 



ILLINOIS ENGINEERING EXPERIMENT STATION 



Applying the relations between stress and strain to a bar in simple 
tension, the following relation between Poisson's ratio and the modulus 

of elasticity is established. Both — and E are to be determined from 

m 

tests in simple tension or compression. 

/x(3A + 2/x) 



E = 



1 



A+/. 
X 



m 2(A+/>i) 
G = the shearing modulus of elasticity = fi. 



(7) 
(8) 



2 \m+l/ 



E 



The values of A and u may now be established in terms of E and 



X = 



Em 



(w+l) {m—2) 
Em 



2(w+l) 
Adding the second and third equations of (6) 

^2 + ^''3 = 2A A + 2/x (c2 4" ^3) 
= 2A AH-2/x(A-ei) 

0-2 + 0-3 + 2/xei 



.(9) 

J_ 

m 
(10) 

(11) 



2(A+/x) 



A A = - (0-2 + o-3+2/xei) 



cri-2/Aei = - (0-2 + 0-3) 



/xei 



\A + i^/ 



O"! = - (0-2 + 0-3) +££1 



W 



Similarly 0-2 = - (o-i -f- 0-3) +££2 



0-3 



1 



w 



(o-i + 0-2) +££. 



(12) 



BECKER STEEL UNDER BIAXIAL LOADING 61 

Eearrangement of Eq. (12) gives 



m 

E€2=(T-2 (o-i+O-s) 

m 

Ee-s =as— — (o-i + cr-i) 
m 



(13) 



These are the three fundamental equations connecting stress and 
strain, ^e^, £'c2 and £'€3 are called by various writers the reduced stresses, 
the true stresses, or the ideal stresses. 

2. Stresses and Strains in a Thin Tube. — For bodies of cylindrical 
form it is convenient to use cylindrical coordinates r, and z instead of 
the rectangular system x, y, z. The z coordinate is measured parallel to 
the axis of the tube, r denotes the radial distance from the axis, and 6 the 
angle of an axial plane from some chosen initial plane. 

Denoting by u, v and w the displacement components, as before 
{u radial, w axial and v perpendicular to a radius r) the three strain 
components e^, e^, e^ are given by the relations 

6u 1 dv u dw 

^r=— ,t^=--^ + -,ez= — (14) 

or r dv r oz 

and the corresponding stress components are given by the equations 

(Tr = AA + 2/ier I 

a^ = AA + V^A (15) 

cTz — AA + 2/.iezJ 

Expressions for the shearing strain and stress components may be 
deduced, but they are not needed in the present investigation. 

In the case of a hollow cylinder under internal pressure, conditions 
of sjnnmetry require that the displacement v shall be zero; hence the 

u 
expression for e^ in (14) reduces to e^ = - , Furthermore, it is per- 

r 

missible in the case under consideration to assume a condition of plane 
strain, in which all points in a cross section of the cylinder experience 
the same displacement w in the z direction. With this assumption, 
w = az, where a is a constant, and therefore €z = cl- With these simplify- 
ing assumptions, the expression for the dilatation takes the form 

d/u. u 

6r r 

If now general expressions for the displacements u and w are found, 
eq. (14) will give tJbe strain components and eq. (15) the stress com- 



62 ILLINOIS ENGINEERING EXPERIMENT STATION 

ponents. The conditions of equilibrium must, however, be satisfied, that 
is relations analogous to (4) must be introduced. It is possible, however, 
to eliminate the stress components by the aid of eq. (6) or eq. (15) 
and thus to express the equilibrium conditions in terms of displacements 
only. Thus from (4) and (3) may be derived the relation 

(X + 2^)— -2m(- —' =0 (16) 

with two similar; and in cylindrical coordinates a similar process leads 
to the relation 

d A /I 6^3 dwoX 
(X+2.)--2.(----^) = (17) 

In the case under consideration the rotation components wo and tog 
must be zero (wg may have a small finite value at the extreme ends 
of the tube), hence (17) reduces to 

d / du u \ 

^'+'^^d^r + ;-^V=' ■ ^''^ 

Integration of this equation leads to the following relation for the 
displacement : 

u = Cr +- 
r 

in which C and D are constant. 

Introducing this expression for u in the expression for €r, c^ and A, 

the result is 

du ^ D 

or r^ 

u ^ D 

eg= -=C + - 



r r 



A=2C -\-a 
Hence the relations (15) become 

o-r =A(2C + a)+2/a 



(-°) 



2C(A + /x) -2/x- + Aa (20) 

^2 



o-^A=(2C + fl) + 2/x('c+^) 



= 2C(A + ^) + 2/x - + A a (21) 

r^ 

o-z=X(2C + a) + 2/>ta 

= 2C A + (A + 2fM)a (22) 



BECKER — STEEL UNDER BIAXIAL LOADING 63 

To determine the constants C and D, we have the conditions 
o-r = —Pi the internal pressure, when r = i\, the internal radius 
o-r — —Po the external pressure, when r = r^, the external radius 

From (20) 

D 

-/^i = 2C(A+/x)-2/x-+Aa 

-^> = 2C(A+^t)-2^- + Aa 
whence 

2 2 

2^D = {p,-po) 4^, (23) 

To /I 

2C(X+.) = ^i^4=^^-Aa (24) 

To fl 

Equations (23) and (24) may be written 

2fxD = T ■(23a) 

2C(A + /x) =S — \a (24a) 



Pi^i'-Po^ 



It will be observed that S = \ ; — srives the mean intensity 

' ' 1 

of tensile stress in a cross-section of a closed tube due to the internal 
fluid pressure p-^^, with external pressure Pq. In the test the axial stress 
was in part applied by the testing machine; hence its value may be 
denoted by JcS, where A; is a constant that becomes equal to 1 when the 
axial stress is one-half of the hoop tension. In the test an axial stress 
was applied by the testing machine and this must be added to the axial 
stress due to internal pressure. Hence the total axial stress may be 
taken as kS. Putting kS for ctz in (22) and combining with (24a), we 
have two equations for the determination of a and C, namely : 
kS = 2\C+ {X + 2fi)a 

2C (\-hfx)=S — Xa 

From these the following results are readily obtained : 

"=I{'-1) ^''^ 

C=-§\l--il+k)] (26) 



Also from (23a) 



ZP=f4^ (27) 



The strain component e^ is now foimd. 



(28) 



64 ILLINOIS ENGINEERING EXPERIMENT STATION 

The value of e^ at the outer surface of the cylinder, where the strain 
was measured, is found by taking r = r^. Substituting now the proper 
expressions for ^S" and T, and putting r^r^, (28) becomes 

1 f,_,n^T _1 (,+,)!+(,,_,„) -;i^l±^l ..(29) 
E L Yo^—ri^ \_ m J r(?—r^ m J 

Since 'p^ is small compared with 'p^, the terms p^r^ — p^r^ and 
{Px — Po) ^2 ^9,y be considered equal. With this approximation (29) 
becomes 

If we consider a closed cylindrical tube with internal hydrostatic 
pressure p^ and external pressure Pq, the net load producing axial tension 
is 

and the area of the cross section of the tube is 

7r(V — V) 
Denoting the load by P and the area by A, we have 

pjr^-por^ ^P 

^'^^-h('-^) (3^) 

-) (32) 

In the test the axial load applied was hP = L; hence 

, , L (2m— k\ ,„ . 

r-f=^) m 

\ km J 
The corresponding values of Ee (reduced stresses) are 

£(e,)„=^?^* (35) 

A km 

L k m -2 , . 

E^z = - —- [Sb) 

A km 



e.=a = ^ik 



L 
EA 



BKCKKn — STKKL rXDKli 15IAXIA1. LOADING 65 

The actual sti-esscs are 



kp 


L 

~ A 


T 




yi 





111 the preceding discussion the results have been obtained in terms 
of /^i and /^(, the ahsohite internal and external fluid pressures. Evi- 
dently //„ is the pressure of the atmosphere, in the test the internal 
pressure p^ was measured by the gage, and no account was taken of the 
external pressure W- This procedure is justified by the following results: 

Let // = y>i — /y,) = internal gage pressure. 

^ pj,^ -p.ro'' p^r,^-poro^-p^r^^-p.r,^ 



Phen 



■rr ri?-—r^ 



and = {py-p^) — : "^ 



ro'-n^ p'r,^ 



r,f- 


— r 


1 




P' 


r„ 


1y 




.. 9 




o 



2 — ^2 



y 2y 2 

T = iP:- p.) -lr\ 
From (38) the hoop tension at the outer surface is • 

Since ]}^, is entirely negligible in comparison with S, we may take 

2L 

{cTg). = 2S = ~- (40) 

k A 

For the corresponding stress at the inner surface, we have 

X (r ^4-r ^) 2L 
(.,). = 5+ - =/ ^--^ =25+/>'= — +p' (41) 



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VITA 

The candidate was born October 6, 1877, in Evansville, Indiana. 
He received his elementary and high school education in the schools of 
that city, graduating from the high school in 1894. From 1894 to 1899 
he was employed in the blacksmith and fitting departments of a plow 
works in P>ansville. 

In the fall of 1899 ^^ entered the Engineering Department of the 
University of Michigan, receiving the degree of B. S. in M. E. in 1903, 
and the M. E. degree in 1907. During the last year of attendance he 
was assistant to Professor M. E. Cooley. From June 1903 to Septem- 
ber 1904 he was with the Kalamazoo Gas Co., Kalamazoo, Michigan, 
as Assistant Superintendent in charge of the enlargement and remodel- 
ing of the plant. 

In September 1904 he went to the University of North Dakota as 
instructor in Mechanical Engineering, and was advanced to Assistant 
Professor in 1906. One year later (1907) he was made Professor of 
Applied Mathematics, which position he still holds. During the past 
year he has been on leave of absence to carry on graduate work at the 
University of Illinois. 

He is a member of Sigma Xi, the Society for the Promotion of 
Engineering Education, and of the American Society of Mechanical 
Engineers. 



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